Foreword_NN1 :_: Alan_NP1 Simpson_NP1 's_GE tribute_NN1 to_II his_APPGE co-author_NN1 My_APPGE co-author_NN1 and_CC friend_NN1 Professor_NNB A._NP1 R._NP1 Collar_NP1 ,_, C.B.E._NNA ,_, LL.D._NP1 ,_, F.R.S._NPD2 ,_, Emeritus_JJ Professor_NN1 of_IO Aeronautical_JJ Engineering_NN1 in_II the_AT University_NN1 of_IO Bristol_NP1 ,_, distinguished_JJ aeronautical_JJ scientist_NN1 and_CC former_DA Vice-Chancellor_NN1 of_IO that_DD1 University_NN1 ,_, died_VVD at_II his_APPGE home_NN1 in_II Bristol_NP1 on_II 12th_MD February_NPM1 1986_MC ,_, just_RR ten_MC days_NNT2 before_II his_APPGE 78th_MD birthday_NN1 ,_, and_CC only_RR a_AT1 short_JJ time_NNT1 before_II our_APPGE book_NN1 was_VBDZ published_VVN ._. 
Roderick_NP1 's_GE eminence_NN1 as_II an_AT1 engineering_NN1 scientist_NN1 and_CC scholar_NN1 is_VBZ well_RR known_VVN ._. 
However_RR ,_, he_PPHS1 was_VBDZ also_RR a_AT1 man_NN1 of_IO exceptional_JJ personal_JJ warmth_NN1 and_CC charm_NN1 ._. 
He_PPHS1 was_VBDZ ,_, in_II every_AT1 sense_NN1 ,_, a_AT1 gentleman_NN1 and_CC a_AT1 gentle_JJ man_NN1 ._. 
Nevertheless_RR ,_, when_CS the_AT occasion_NN1 demanded_VVD it_PPH1 (_( as_CSA it_PPH1 did_VDD during_II the_AT turbulent_JJ period_NN1 in_II which_DDQ he_PPHS1 became_VVD Vice-Chancellor_NN1 of_IO Bristol_NP1 University_NN1 )_) ,_, he_PPHS1 showed_VVD great_JJ courage_NN1 and_CC firmness_NN1 the_AT latter_DA always_RR in_II the_AT nicest_JJT possible_JJ way_NN1 ._. 
Roderick_NP1 's_GE interests_NN2 were_VBDR wide_JJ ;_; for_REX21 example_REX22 sport_NN1 (_( cricket_NN1 ,_, tennis_NN1 and_CC football_NN1 both_DB2 varieties_NN2 )_) ,_, good_JJ music_NN1 (_( he_PPHS1 played_VVD the_AT violin_NN1 )_) ,_, verse_NN1 (_( he_PPHS1 read_VVD extensively_RR and_CC composed_VVD ,_, when_CS inspired_VVN or_CC provoked_VVD ,_, to_TO match_VVI an_AT1 occasion_NN1 and_CC could_VM recite_VVI extensively_RR from_II memory_NN1 particularly_RR W._NP1 S._NP1 Gilbert_NP1 )_) ,_, games_NN2 and_CC puzzles_NN2 (_( chess_NN1 ,_, cribbage_VV0 ,_, mathematical_JJ conundrums_NN2 and_CC crosswords_NN2 ;_; he_PPHS1 could_VM ,_, at_II his_APPGE zenith_NN1 ,_, do_VD0 The_AT Times_NNT2 Crossword_NP1 any_DD day_NNT1 in_II less_DAR than_CSN twenty_MC minutes_NNT2 )_) ,_, and_CC art_NN1 (_( he_PPHS1 was_VBDZ an_AT1 active_JJ member_NN1 of_IO The_AT Bristol_NP1 Savages_NN2 )_) ._. 
He_PPHS1 was_VBDZ also_RR president_NN1 of_IO a_AT1 local_JJ Gilbert_NP1 and_CC Sullivan_NP1 society_NN1 for_IF many_DA2 years_NNT2 ._. 
He_PPHS1 did_VDD much_DA1 hospital_NN1 work_VVI :_: for_IF several_DA2 years_NNT2 up_II21 to_II22 his_APPGE retirement_NN1 he_PPHS1 had_VHD been_VBN Chairman_NN1 of_IO the_AT South_ND1 West_ND1 Regional_JJ Hospital_NN1 Management_NN1 Board_NN1 and_CC was_VBDZ an_AT1 active_JJ hospital_NN1 visitor_NN1 ._. 
His_APPGE charitable_JJ works_NN were_VBDR extensive_JJ ._. 
He_PPHS1 had_VHD been_VBN President_NN1 of_IO the_AT Bristol_NP1 Rotary_JJ Club_NN1 ,_, chairman_NN1 of_IO governors_NN2 of_IO many_DA2 schools_NN2 and_CC colleges_NN2 (_( notably_RR R.M.C.S._NP1 ,_, Shrivenham_NP1 )_) ,_, and_CC was_VBDZ a_AT1 Liveryman_NN1 of_IO the_AT Guild_NN1 of_IO Air_NN1 Pilots_NN2 and_CC Navigators_NN2 ,_, to_TO mention_VVI but_CCB a_AT1 few_DA2 of_IO his_APPGE activities_NN2 ._. 
He_PPHS1 was_VBDZ the_AT perfect_JJ committee_NN1 chairman_NN1 :_: he_PPHS1 loved_VVD committee_NN1 work_NN1 when_CS it_PPH1 was_VBDZ productive_JJ ,_, and_CC it_PPH1 always_RR was_VBDZ when_RRQ he_PPHS1 was_VBDZ in_II the_AT chair_NN1 ._. 
Innate_JJ within_II him_PPHO1 was_VBDZ the_AT ability_NN1 to_TO put_VVI committee_NN1 members_NN2 at_II their_APPGE ease_NN1 ,_, so_CS21 that_CS22 they_PPHS2 could_VM give_VVI of_IO their_APPGE best_JJT ,_, and_CC he_PPHS1 could_VM abstract_VVI the_AT maximum_JJ amount_NN1 of_IO information_NN1 ._. 
This_DD1 ability_NN1 was_VBDZ evident_JJ in_II his_APPGE many_DA2 years_NNT2 of_IO A.R.C._NP1 Committee_NN1 and_CC Council_NN1 chairmanship_NN1 and_CC in_II his_APPGE chairmanship_NN1 of_IO the_AT university_NN1 committees_NN2 ,_, culminating_VVG in_II his_APPGE Vice-Chancellorship_NN1 ._. 
At_II one_MC1 time_NNT1 he_PPHS1 was_VBDZ on_II 25_MC committees_NN2 ,_, over_II and_CC above_II those_DD2 within_II the_AT University_NN1 ,_, and_CC was_VBDZ chairman_NN1 of_IO many_DA2 of_IO these_DD2 ._. 
At_II the_AT height_NN1 of_IO this_DD1 committee_NN1 and_CC other_JJ administrative_JJ work_NN1 ,_, Roderick_NP1 often_RR expressed_VVD his_APPGE desire_NN1 to_TO return_VVI to_II his_APPGE mathematics_NN1 and_CC ,_, in_RR21 particular_RR22 ,_, to_TO write_VVI a_AT1 sequel_NN1 to_II his_APPGE famous_JJ book_NN1 (_( with_IW R.A._NP1 Frazer_NP1 and_CC W.J._NP1 Duncan_NP1 )_) Elementary_JJ Matrices_NN2 ._. 
I_PPIS1 had_VHD agreed_VVN to_TO be_VBI co-author_NN1 ,_, but_CCB it_PPH1 was_VBDZ not_XX until_CS seven_MC years_NNT2 after_II his_APPGE retirement_NN1 that_CST work_NN1 could_VM begin_VVI ,_, for_IF only_RR then_RT had_VHD his_APPGE committee_NN1 activities_NN2 abated_VVN to_II an_AT1 extent_NN1 which_DDQ allowed_VVD the_AT long_JJ periods_NN2 necessary_JJ for_IF writing_VVG and_CC discussion_NN1 ._. 
It_PPH1 is_VBZ appropriate_JJ to_TO recall_VVI that_DD1 Elementary_JJ Matrices_NN2 (_( 1938_MC )_) was_VBDZ printed_VVN nine_MC times_NNT2 in_II U.K._NP1 ,_, several_DA2 times_NNT2 in_II U.S.A._NP1 ,_, was_VBDZ translated_VVN into_II Russian_JJ and_CC Czech_JJ ,_, and_CC finally_RR republished_VVD in_II hardback_NN1 in_II U.S.A._NP1 in_II 1983_MC ._. 
It_PPH1 was_VBDZ written_VVN following_II a_AT1 period_NN1 of_IO pioneering_JJ work_NN1 on_II the_AT application_NN1 of_IO matrices_NN2 to_II engineering_NN1 problems_NN2 ,_, undertaken_VVN by_II Frazer_NP1 ,_, Duncan_NP1 and_CC Collar_NN1 at_II the_AT N.P.L._NP1 ,_, Teddington_NP1 ,_, in_II the_AT 1930s_MC2 ._. 
Collar_NN1 was_VBDZ certainly_RR the_AT junior_JJ member_NN1 of_IO the_AT partnership_NN1 ,_, but_CCB those_DD2 who_PNQS know_VV0 his_APPGE mathematical_JJ style_NN1 are_VBR well_JJ able_JJ to_TO perceive_VVI his_APPGE many_DA2 areas_NN2 of_IO contribution_NN1 to_II the_AT book_NN1 ._. 
In_II later_JJR years_NNT2 ,_, Roderick_NP1 confided_VVD that_CST he_PPHS1 had_VHD been_VBN far_RG21 from_RG22 happy_JJ with_IW the_AT title_NN1 ,_, for_IF the_AT book_NN1 certainly_RR was_VBDZ not_XX &quot;_" elementary_JJ &quot;_" ._. 
Any_DD sequel_NN1 must_VM be_VBI &quot;_" more_RGR elementary_JJ &quot;_" and_CC tailored_VVN to_II the_AT needs_NN2 of_IO the_AT engineering_NN1 student_NN1 ._. 
This_DD1 is_VBZ not_XX to_TO be_VBI viewed_VVN as_CSA condescending_JJ :_: Roderick_NP1 regarded_VVD himself_PPX1 as_II a_AT1 student_NN1 of_IO the_AT subject_NN1 and_CC said_VVD so_RR on_II many_DA2 occasions_NN2 during_II the_AT writing_NN1 of_IO the_AT new_JJ book_NN1 ._. 
As_CSA work_NN1 began_VVD in_II earnest_JJ on_II the_AT book_NN1 ,_, I_PPIS1 realised_VVD that_CST Roderick_NP1 had_VHD lost_VVN none_PN of_IO the_AT skills_NN2 that_CST I_PPIS1 ,_, as_CSA his_APPGE research_NN1 student_NN1 twenty_MC years_NNT2 before_RT ,_, had_VHD regarded_VVN with_IW awe_NN1 ._. 
But_CCB writing_VVG now_RT was_VBDZ not_XX easy_JJ for_IF him_PPHO1 :_: he_PPHS1 had_VHD begun_VVN to_TO suffer_VVI with_IW arthritis_NN1 in_II the_AT year_NNT1 following_VVG his_APPGE retirement_NN1 ,_, and_CC had_VHD had_VHN to_TO give_VVI up_RP his_APPGE violin_NN1 owing_II21 to_II22 lack_NN1 of_IO flexibility_NN1 of_IO his_APPGE fingers_NN2 ,_, and_CC as_II the_AT condition_NN1 developed_VVD ,_, he_PPHS1 found_VVD writing_VVG more_RRR and_CC more_RGR difficult_JJ ._. 
This_DD1 did_VDD not_XX deter_VVI him_PPHO1 ,_, as_CSA ,_, between_II 1980_MC and_CC 1984_MC ,_, he_PPHS1 completed_VVD over_RG 400_MC pages_NN2 of_IO manuscript_NN1 ._. 
On_II average_NN1 ,_, we_PPIS2 met_VVD for_IF three_MC hours_NNT2 every_AT1 ten_MC days_NNT2 ,_, constantly_RR revising_VVG ,_, exchanging_VVG and_CC criticising_VVG so_CS21 that_CS22 in_II the_AT end_NN1 we_PPIS2 had_VHD both_RR had_VHN a_AT1 hand_NN1 in_II everything_PN1 ._. 
The_AT whole_NN1 must_VM have_VHI been_VBN rewritten_VVN about_RG five_MC times_NNT2 ._. 
Roderick_NP1 was_VBDZ always_RR seeking_VVG a_AT1 better_JJR way_NN1 of_IO putting_VVG things_NN2 in_BCL21 order_BCL22 to_TO remove_VVI all_DB possible_JJ ambiguities_NN2 and_CC to_TO assist_VVI the_AT reader_NN1 to_II the_AT maximum_JJ extent_NN1 :_: if_CS this_DD1 meant_VVD rewriting_VVG 100_MC times_NNT2 he_PPHS1 would_VM gladly_RR do_VDI it_PPH1 ._. 
He_PPHS1 was_VBDZ meticulous_JJ ,_, but_CCB never_RR pedantic_JJ ._. 
Roderick_NP1 's_GE last_MD ,_, and_CC possibly_RR finest_JJT ,_, contributions_NN2 to_II this_DD1 book_NN1 were_VBDR the_AT Theorems_NN2 XII_MC and_CC XV_MC of_IO Chapter_NN1 1_MC1 which_DDQ were_VBDR completed_VVN in_II 1985_MC shortly_RR after_II the_AT serious_JJ fall_NN1 which_DDQ perhaps_RR precipitated_VVD the_AT leukaemia_NN1 from_II which_DDQ he_PPHS1 failed_VVD to_TO recover_VVI ,_, despite_II repeated_JJ blood_NN1 transfusions_NN2 ._. 
At_II this_DD1 stage_NN1 ,_, the_AT arthritis_NN1 in_II his_APPGE fingers_NN2 was_VBDZ scarcely_RR bearable_JJ and_CC writing_VVG extremely_RR painful_JJ ._. 
But_CCB the_AT exceptional_JJ clarity_NN1 of_IO his_APPGE thought_NN1 processes_NN2 was_VBDZ evidently_RR still_RR totally_RR unimpaired_JJ ._. 
We_PPIS2 last_VV0 met_VVN on_II the_AT Sunday_NPD1 before_II his_APPGE death_NN1 ._. 
Although_CS very_RG weak_JJ ,_, he_PPHS1 was_VBDZ anxious_JJ to_TO discuss_VVI technical_JJ matters_NN2 ,_, among_II which_DDQ were_VBDR possible_JJ revisions_NN2 to_II Chapter_NN1 6_MC ._. 
He_PPHS1 had_VHD ,_, a_AT1 few_DA2 weeks_NNT2 before_RT ,_, completed_VVD an_AT1 excellent_JJ set_NN1 of_IO autobiographical_JJ notes_NN2 and_CC ,_, characteristically_RR ,_, a_AT1 sonnet_NN1 to_II his_APPGE wife_NN1 ,_, Bobbie_NP1 ,_, on_II her_APPGE birthday_NN1 ._. 
Roderick_NP1 will_VM be_VBI sadly_RR missed_VVN in_II every_AT1 circle_NN1 in_II which_DDQ he_PPHS1 moved_VVD ,_, but_CCB most_DAT of_IO all_DB ,_, of_RR21 course_RR22 ,_, by_II Bobbie_NP1 ,_, whose_DDQGE devotion_NN1 and_CC fortitude_NN1 was_VBDZ his_APPGE greatest_JJT comfort_NN1 in_II his_APPGE final_JJ illness_NN1 ,_, and_CC whose_DDQGE forbearance_NN1 over_II many_DA2 years_NNT2 enabled_VVD Roderick_NP1 to_TO spend_VVI such_DA a_AT1 great_JJ amount_NN1 of_IO their_APPGE valuable_JJ time_NNT1 on_II the_AT book_NN1 ._. 
A.S._NP1 ,_, February_NPM1 1986_MC ._. 
INTRODUCTION_NN1 In_II this_DD1 text_NN1 ,_, which_DDQ is_VBZ aimed_VVN at_II senior_JJ undergraduate_NN1 students_NN2 of_IO engineering_NN1 ,_, the_AT subject_NN1 of_IO matrices_NN2 is_VBZ described_VVN and_CC developed_VVN before_II being_VBG applied_VVN to_II some_DD of_IO the_AT problems_NN2 of_IO engineering_NN1 dynamics_NN ._. 
The_AT matrix_NN1 theory_NN1 is_VBZ presented_VVN in_II classical_JJ algebraic_JJ form_NN1 with_IW no_AT recourse_NN1 to_II the_AT notions_NN2 and_CC nomenclature_NN1 of_IO vector_NN1 space_NN1 theory_NN1 ._. 
In_II this_DD1 respect_NN1 ,_, the_AT book_NN1 is_VBZ a_AT1 sequel_NN1 to_II the_AT earlier_JJR work_NN1 Elementary_JJ Matrices_NN2 by_II Frazer_NP1 ,_, Duncan_NP1 and_CC Collar_NN1 ,_, a_AT1 book_NN1 which_DDQ ,_, by_II presenting_VVG problems_NN2 in_II a_AT1 form_NN1 which_DDQ could_VM be_VBI assimilated_VVN by_II computers_NN2 ,_, stimulated_VVD the_AT growth_NN1 of_IO the_AT latter_DA ._. 
In_II the_AT present_JJ book_NN1 ,_, the_AT subject_NN1 is_VBZ brought_VVN up_RP to_II date_NN1 and_CC related_VVN to_II modern_JJ computer_NN1 methods_NN2 ._. 
The_AT text_NN1 is_VBZ punctuated_VVN throughout_RL by_II numerical_JJ examples_NN2 ,_, most_RGT usually_RR based_VVN on_II matrices_NN2 of_IO small_JJ order_NN1 ._. 
The_AT treatment_NN1 of_IO engineering_NN1 dynamics_NN is_VBZ almost_RR invariably_RR linear_JJ ,_, although_CS examples_NN2 of_IO simple_JJ non-linear_JJ formulations_NN2 are_VBR provided_VVN ._. 
Chapter_NN1 1_MC1 contains_VVZ definitions_NN2 of_IO the_AT basic_JJ laws_NN2 of_IO matrix_NN1 manipulation_NN1 and_CC of_IO matrix_NN1 types_NN2 ._. 
Matrix_NN1 calculus_NN1 is_VBZ introduced_VVN via_II the_AT Sylvester_NP1 expansion_NN1 theorem_NN1 and_CC considerable_JJ emphasis_NN1 is_VBZ placed_VVN on_II the_AT matrix_NN1 eigenvalue_NN1 problem_NN1 ._. 
A_AT1 useful_JJ collection_NN1 of_IO theorems_NN2 and_CC proofs_NN2 is_VBZ presented_VVN at_II the_AT end_NN1 of_IO the_AT chapter_NN1 :_: among_II these_DD2 is_VBZ what_DDQ is_VBZ thought_VVN to_TO be_VBI a_AT1 novel_JJ proof_NN1 of_IO Sylvester_NP1 's_GE law_NN1 of_IO degeneracy_NN1 ._. 
Chapter_NN1 2_MC is_VBZ concerned_JJ with_IW numerical_JJ methods_NN2 :_: it_PPH1 divides_VVZ into_II two_MC major_JJ parts_NN2 ,_, namely_REX methods_NN2 of_IO reciprocation_NN1 ,_, triangulation_NN1 and_CC solution_NN1 of_IO linear_JJ algebraic_JJ equations_NN2 ,_, and_CC methods_NN2 for_IF the_AT solution_NN1 of_IO eigenvalue_NN1 problems_NN2 ._. 
Both_DB2 parts_NN2 address_VV0 many_DA2 alternative_JJ procedures_NN2 ,_, all_DB of_IO which_DDQ are_VBR exemplified_VVN most_RGT often_RR by_II using_VVG matrices_NN2 of_IO order_NN1 4_MC ._. 
Chapter_NN1 3_MC concerns_VVZ neighbour_NN1 systems_NN2 ,_, a_AT1 subject_NN1 of_IO great_JJ importance_NN1 in_II engineering_NN1 dynamics_NN ._. 
The_AT Collar-Jahn_NP1 method_NN1 ,_, along_II21 with_II22 several_DA2 new_JJ variants_NN2 ,_, is_VBZ presented_VVN ._. 
The_AT difficult_JJ problems_NN2 associated_VVN with_IW adjacent_JJ or_CC confluent_JJ eigenvalues_NN2 are_VBR outlined_VVN and_CC methods_NN2 of_IO solution_NN1 developed_VVN and_CC discussed_VVN ._. 
The_AT chapter_NN1 is_VBZ thought_VVN to_TO be_VBI the_AT first_MD on_II this_DD1 subject_NN1 in_II a_AT1 text_NN1 of_IO this_DD1 type_NN1 ._. 
Fundamental_JJ concepts_NN2 of_IO dynamical_JJ systems_NN2 are_VBR introduced_VVN in_II the_AT simplest_JJT of_IO terms_NN2 ,_, in_II Chapter_NN1 4_MC ._. 
Formulations_NN2 based_VVN on_II the_AT principle_NN1 of_IO virtual_JJ work_NN1 are_VBR given_VVN emphasis_NN1 :_: conservative_JJ and_CC non-conservative_JJ actions_NN2 are_VBR carefully_RR delineated_VVN and_CC discussed_VVN in_II the_AT context_NN1 of_IO discrete_JJ systems_NN2 ._. 
The_AT important_JJ notion_NN1 of_IO semi-rigidity_NN1 is_VBZ introduced_VVN ._. 
The_AT ideas_NN2 of_IO Chapter_NN1 4_MC are_VBR carried_VVN forward_RL in_II Chapter_NN1 5_MC via_II a_AT1 proof_NN1 of_IO Hamilton_NP1 's_GE principle_NN1 and_CC the_AT associated_JJ proof_NN1 of_IO the_AT Lagrange_NN1 equations_NN2 ._. 
Conservative_NN1 and_CC non-conservative_JJ actions_NN2 are_VBR identified_VVN by_II the_AT form_NN1 of_IO the_AT matrices_NN2 appearing_VVG as_II coefficient_NN1 arrays_NN2 in_II the_AT equations_NN2 of_IO motion_NN1 ._. 
A_AT1 general_JJ treatment_NN1 of_IO the_AT linear_JJ conservative_JJ dynamical_JJ system_NN1 is_VBZ given_VVN ,_, including_II calculation_NN1 of_IO natural_JJ frequencies_NN2 and_CC normal_JJ modes_NN2 ,_, and_CC topics_NN2 such_II21 as_II22 removal_NN1 of_IO zero_NN1 frequency_NN1 roots_NN2 ._. 
An_AT1 outline_NN1 of_IO structural_JJ damping_NN1 modelling_NN1 is_VBZ also_RR given_VVN ._. 
Chapter_NN1 6_MC deals_NN2 in_II general_JJ terms_NN2 with_IW the_AT solution_NN1 of_IO sets_NN2 of_IO linear_JJ differential_JJ equations_NN2 with_IW constant_JJ coefficients_NN2 ._. 
The_AT concept_NN1 of_IO the_AT matrix_NN1 Laplace_NP1 transform_VV0 is_VBZ introduced_VVN ._. 
In_II the_AT second_MD part_NN1 of_IO his_APPGE chapter_NN1 ,_, the_AT subject_NN1 of_IO linear_JJ stability_NN1 theory_NN1 is_VBZ discussed_VVN ,_, leading_VVG from_II the_AT fundamental_JJ encirclement_NN1 theorem_NN1 of_IO Cauchy_NP1 ,_, via_II the_AT criteria_NN2 of_IO Leonhard_NP1 and_CC Routh_NP1 ,_, to_II the_AT Poincar-Liapunov_NP1 method_NN1 ._. 
An_AT1 inverse_JJ method_NN1 for_IF the_AT calculation_NN1 of_IO stability_NN1 boundaries_NN2 is_VBZ also_RR discussed_VVN ._. 
Chapter_NN1 7_MC gives_VVZ an_AT1 extended_JJ treatment_NN1 of_IO the_AT dynamics_NN problems_NN2 of_IO continuous_JJ linear_JJ systems_NN2 by_II use_NN1 of_IO the_AT semi-rigid_JJ method_NN1 in_II matrix_NN1 form_NN1 ._. 
Emphasis_NN1 is_VBZ placed_VVN on_II one-dimensional_JJ systems_NN2 ,_, but_CCB flat_JJ plate_NN1 problems_NN2 are_VBR also_RR discussed_VVN ._. 
Exact_JJ formulations_NN2 based_VVN on_II &quot;_" dynamic_JJ stiffness_NN1 &quot;_" or_CC &quot;_" receptance_NN1 &quot;_" are_VBR highlighted_VVN ,_, and_CC the_AT concept_NN1 of_IO &quot;_" finite_JJ element_NN1 &quot;_" is_VBZ introduced_VVN as_II a_AT1 natural_JJ extension_NN1 of_IO the_AT semi-rigid_JJ method_NN1 ._. 
Finally_RR ,_, Chapter_NN1 8_MC deals_NN2 with_IW the_AT dynamics_NN of_IO composite_JJ systems_NN2 in_II a_AT1 manner_NN1 which_DDQ owes_VVZ much_RR to_II the_AT pioneering_JJ work_NN1 of_IO Kron_NN1 ._. 
The_AT finite_JJ element_NN1 and_CC dynamic_JJ stiffness_NN1 methods_NN2 are_VBR introduced_VVN and_CC exemplified_VVN in_II simple_JJ cases_NN2 ._. 
The_AT importance_NN1 of_IO the_AT connexion_NN1 graph_NN1 is_VBZ emphasised_VVN ._. 
The_AT chapter_NN1 ends_VVZ with_IW an_AT1 extensive_JJ discussion_NN1 of_IO Kron_NP1 's_GE eigenvalue_NN1 method_NN1 followed_VVN by_II proofs_NN2 of_IO the_AT Wittrick-Williams_NP1 and_CC Simpson_NP1 counting_NN1 algorithms_NN2 ._. 
The_AT book_NN1 will_VM be_VBI accompanied_VVN by_II a_AT1 &quot;_" computational_JJ adjunct_NN1 &quot;_" in_II which_DDQ a_AT1 number_NN1 of_IO BASIC_NP1 programs_NN2 for_IF use_NN1 on_II home_NN1 computers_NN2 will_VM be_VBI presented_VVN ._. 
These_DD2 programs_NN2 ,_, which_DDQ will_VM be_VBI available_JJ on_II disc_NN1 ,_, relate_VV0 to_II specific_JJ topics_NN2 covered_VVN in_II the_AT main_JJ text_NN1 and_CC may_VM be_VBI used_VVN to_TO verify_VVI calculations_NN2 performed_VVD therein_RR ._. 
Some_DD of_IO the_AT programs_NN2 ,_, however_RR ,_, are_VBR of_IO the_AT general-purpose_JJ type_NN1 ._. 
Properties_NN2 of_IO Matrices_NN2 1.1_MC ASSUMPTIONS_NN2 ,_, CONVENTIONS_NN2 ;_; TRANSPOSITION_NP1 AND_CC RELATED_JJ DEFINITIONS_NN2 In_II this_DD1 chapter_NN1 we_PPIS2 shall_VM discuss_VVI certain_JJ characteristics_NN2 and_CC properties_NN2 of_IO matrices_NN2 which_DDQ are_VBR of_IO particular_JJ importance_NN1 in_II dynamical_JJ studies_NN2 of_IO systems_NN2 with_IW multiple_JJ degrees_NN2 of_IO freedom_NN1 ._. 
We_PPIS2 assume_VV0 the_AT reader_NN1 to_TO be_VBI familiar_JJ with_IW the_AT notation_NN1 of_IO matrices_NN2 and_CC with_IW the_AT rules_NN2 of_IO combination_NN1 :_: addition_NN1 ,_, multiplication_NN1 ,_, etc._RA ,_, and_CC with_IW conformability_NN1 and_CC non-permutability_NN1 ._. 
We_PPIS2 also_RR assume_VV0 familiarity_NN1 with_IW the_AT elementary_JJ rules_NN2 for_IF combination_NN1 and_CC evaluation_NN1 of_IO determinants_NN2 ._. 
In_RR21 general_RR22 ,_, matrices_NN2 are_VBR rectangular_JJ ;_; but_CCB of_IO particular_JJ importance_NN1 are_VBR matrices_NN2 having_VHG equal_JJ numbers_NN2 of_IO rows_NN2 and_CC columns_NN2 i.e._REX square_JJ matrices_NN2 and_CC those_DD2 having_VHG only_RR a_AT1 single_JJ row_NN1 or_CC column_NN1 ,_, which_DDQ we_PPIS2 call_VV0 vectors_NN2 ;_; both_DB2 types_NN2 are_VBR special_JJ forms_NN2 of_IO rectangular_JJ matrices_NN2 ._. 
Normally_RR ,_, we_PPIS2 shall_VM use_VVI capital_JJ bold_JJ type_NN1 ,_, e.g._REX A_ZZ1 ,_, to_TO denote_VVI square_JJ and_CC rectangular_JJ matrices_NN2 having_VHG two_MC or_CC more_DAR rows_NN2 and_CC columns_NN2 ,_, and_CC lower_JJR case_NN1 bold_JJ type_NN1 ,_, e.g._REX x_ZZ1 ,_, to_TO denote_VVI vectors_NN2 ._. 
If_CS a_AT1 matrix_NN1 A_ZZ1 has_VHZ m_MC rows_NN2 and_CC n_ZZ1 columns_NN2 ,_, i.e._REX it_PPH1 is_VBZ of_IO order_NN1 (_( m_ZZ1 n_ZZ1 )_) ,_, its_APPGE typical_JJ element_NN1 &lsqb;_( formula_NN1 &rsqb;_) lies_VVZ at_II the_AT intersection_NN1 of_IO the_AT ith_MD row_NN1 and_CC jth_MD column_NN1 (_( &lsqb;_( formula_NN1 &rsqb;_) )_) ._. 
If_CS we_PPIS2 form_VV0 the_AT matrix_NN1 &lsqb;_( formula_NN1 &rsqb;_) by_II writing_VVG columns_NN2 for_IF rows_NN2 in_II order_NN1 ,_, the_AT new_JJ matrix_NN1 ,_, of_IO order_NN1 (_( n_ZZ1 m_ZZ1 )_) is_VBZ called_VVN the_AT transpose_VV0 of_IO A_ZZ1 and_CC is_VBZ denoted_VVN by_II &lsqb;_( formula_NN1 &rsqb;_) ._. 
For_REX21 example_REX22 ,_, &lsqb;_( formula_NN1 &rsqb;_) ._. 
The_AT process_NN1 is_VBZ clearly_RR one_MC1 of_IO reflection_NN1 in_II the_AT diagonal_JJ containing_VVG the_AT terms_NN2 &lsqb;_( formula_NN1 &rsqb;_) ,_, which_DDQ is_VBZ described_VVN as_II the_AT principal_JJ diagonal_JJ ._. 
It_PPH1 is_VBZ of_IO particular_JJ importance_NN1 in_II square_JJ matrices_NN2 ._. 
The_AT terms_NN2 &lsqb;_( formula_NN1 &rsqb;_) in_II A_ZZ1 form_VV0 the_AT superdiagonal_JJ and_CC &lsqb;_( formula_NN1 &rsqb;_) the_AT infradiagonal_JJ ,_, while_CS for_IF a_AT1 square_JJ matrix_NN1 of_IO order_NN1 (_( n_ZZ1 n_ZZ1 )_) or_CC more_RGR simply_RR ,_, of_IO order_NN1 n_ZZ1 the_AT elements_NN2 &lsqb;_( formula_NN1 &rsqb;_) ,_, form_VV0 the_AT secondary_JJ diagonal_JJ ._. 
An_AT1 important_JJ property_NN1 of_IO a_AT1 square_JJ matrix_NN1 is_VBZ its_APPGE trace_NN1 ;_; this_DD1 is_VBZ the_AT sum_NN1 of_IO all_DB the_AT elements_NN2 in_II the_AT principal_JJ diagonal_JJ ._. 
Wherever_RRQV it_PPH1 saves_VVZ space_NN1 ,_, we_PPIS2 shall_VM always_RR write_VVI a_AT1 column_NN1 vector_NN1 x_ZZ1 as_CSA &lsqb;_( formula_NN1 &rsqb;_) the_AT braces_NN2 denote_VV0 that_CST this_DD1 is_VBZ to_TO be_VBI read_VVN as_II a_AT1 column_NN1 ._. 
Since_CS in_II dynamics_NN ,_, we_PPIS2 are_VBR usually_RR more_RGR often_RR concerned_JJ with_IW columns_NN2 than_CSN with_IW rows_NN2 ,_, we_PPIS2 shall_VM write_VVI a_AT1 row_NN1 vector_NN1 as_II the_AT transpose_VV0 of_IO a_AT1 column_NN1 vector_NN1 ,_, e.g._REX &lsqb;_( formula_NN1 &rsqb;_) ._. 
1.2_MC SYMMETRIC_JJ AND_CC SKEW-SYMMETRIC_JJ MATRICES_NN2 If_CS &lsqb;_( formula_NN1 &rsqb;_) ,_, the_AT matrices_NN2 ,_, which_DDQ are_VBR necessarily_RR square_JJ ,_, are_VBR termed_VVN symmetric_JJ ;_; if_CS &lsqb;_( formula_NN1 &rsqb;_) ,_, they_PPHS2 are_VBR said_VVN to_TO be_VBI skew-symmetric_JJ :_: in_II this_DD1 case_NN1 ,_, since_CS &lsqb;_( formula_NN1 &rsqb;_) =_FO &lsqb;_( formula_NN1 &rsqb;_) ,_, the_AT principal_JJ diagonal_JJ has_VHZ zero_MC elements_NN2 ._. 
A_AT1 square_JJ matrix_NN1 having_VHG zeros_NN2 everywhere_RL except_CS in_II the_AT principal_JJ diagonal_JJ is_VBZ called_VVN a_AT1 diagonal_JJ matrix_NN1 and_CC is_VBZ clearly_RR symmetric_JJ ._. 
The_AT unit_NN1 matrix_NN1 I_ZZ1 is_VBZ a_AT1 diagonal_JJ matrix_NN1 of_IO unit_NN1 elements_NN2 ;_; a_AT1 matrix_NN1 having_VHG all_RR its_APPGE elements_NN2 zero_NN1 is_VBZ said_VVN to_TO be_VBI null_JJ and_CC is_VBZ written_VVN 0_MC ._. 
It_PPH1 is_VBZ possible_JJ to_TO express_VVI any_DD square_JJ matrix_NN1 as_II the_AT sum_NN1 of_IO a_AT1 symmetric_JJ and_CC a_AT1 skew-symmetric_JJ matrix_NN1 ;_; this_DD1 is_VBZ shown_VVN by_II the_AT identity_NN1 &lsqb;_( formula_NN1 &rsqb;_) ._. 
The_AT first_MD term_NN1 is_VBZ unaltered_JJ by_II transposition_NN1 ;_; the_AT second_MD changes_NN2 sign._NNU 1.3_MC REVERSAL_NN1 OF_IO ORDER_NN1 ON_II TRANSPOSITION_NP1 If_CS we_PPIS2 have_VH0 a_AT1 matrix_NN1 product_NN1 A_ZZ1 =_FO BC_RA ,_, where_CS B_ZZ1 is_VBZ of_IO order_NN1 (_( m_ZZ1 r_ZZ1 )_) and_CC C_ZZ1 of_IO order_NN1 (_( r_ZZ1 n_ZZ1 )_) ,_, then_RT A_ZZ1 is_VBZ of_IO order_NN1 (_( m_ZZ1 n_ZZ1 )_) ._. 
If_CS &lsqb;_( formula_NN1 &rsqb;_) ,_, B_ZZ1 and_CC C_ZZ1 are_VBR conformable_JJ only_RR in_II that_DD1 order_NN1 ._. 
The_AT typical_JJ product_NN1 element_NN1 is_VBZ &lsqb;_( formula_NN1 &rsqb;_) ._. 
Transpose_VV0 B_ZZ1 and_CC C_ZZ1 ;_; Then_RT &lsqb;_( formula_NN1 &rsqb;_) are_VBR of_IO orders_NN2 (_( r_ZZ1 m_ZZ1 )_) ,_, (_( n_ZZ1 r_ZZ1 )_) respectively_RR ,_, and_CC are_VBR therefore_RR conformable_JJ only_RR in_II the_AT order_NN1 &lsqb;_( formula_NN1 &rsqb;_) ._. 
Moreover_RR &lsqb;_( formula_NN1 &rsqb;_) ,_, or_CC &lsqb;_( formula_NN1 &rsqb;_) ._. 
Hence_RR transposition_NN1 of_IO a_AT1 matrix_NN1 product_NN1 requires_VVZ reversal_NN1 of_IO the_AT order_NN1 of_IO the_AT matrices_NN2 ._. 
By_II an_AT1 obvious_JJ extension_NN1 ,_, if_CS A_ZZ1 =_FO BCD_NP1 ,_, then_RT &lsqb;_( formula_NN1 &rsqb;_) ,_, and_RR31 so_RR32 on_RR33 ._. 
In_II31 view_II32 of_II33 the_AT formulae_NN2 (_( 1_MC1 )_) and_CC (_( 2_MC )_) it_PPH1 is_VBZ clear_JJ that_CST these_DD2 results_NN2 hold_VV0 whether_CSW31 or_CSW32 not_CSW33 B_ZZ1 ,_, C_ZZ1 are_VBR conformable_JJ both_RR in_II the_AT order_NN1 BC_RA and_CC in_II the_AT order_NN1 CB_NN1 ,_, including_II the_AT case_NN1 where_CS both_DB2 are_VBR square_JJ ._. 
For_IF the_AT unit_NN1 and_CC null_JJ matrices_NN2 ,_, (_( 1_MC1 )_) implies_VVZ &lsqb;_( formula_NN1 &rsqb;_) ._. 
1.4_MC ._. 
PARTITIONING_NN1 OF_IO MATRICES_NN2 Any_DD matrix_NN1 may_VM be_VBI partitioned_VVN ,_, by_II drawing_VVG dotted_JJ lines_NN2 between_II adjacent_JJ columns_NN2 and_CC adjacent_JJ rows_NN2 ,_, into_II submatrices_NN2 ._. 
We_PPIS2 do_VD0 this_DD1 ,_, notionally_RR ,_, in_II any_DD matrix_NN1 multiplication_NN1 :_: if_CS we_PPIS2 are_VBR performing_VVG the_AT operation_NN1 BC_NP1 =_FO A_ZZ1 then_RT we_PPIS2 isolate_VV0 a_AT1 row_NN1 of_IO B_ZZ1 and_CC a_AT1 column_NN1 of_IO C_NP1 respectively_RR (_( 1_MC1 n_ZZ1 )_) and_CC (_( n_ZZ1 1_MC1 )_) submatrices_VVZ to_TO produce_VVI an_AT1 element_NN1 i.e._REX a_AT1 (_( 1_MC1 1_MC1 )_) submatrix_NN1 of_IO A._NP1 Similarly_RR ,_, we_PPIS2 could_VM ,_, for_REX21 example_REX22 ,_, isolate_VV0 the_AT first_MD three_MC rows_NN2 of_IO B_ZZ1 and_CC the_AT first_MD four_MC columns_NN2 of_IO C_ZZ1 and_CC multiply_VV0 these_DD2 submatrices_NN2 ,_, of_IO order_NN1 (_( 3_MC n_ZZ1 )_) and_CC (_( n_ZZ1 4_MC )_) ,_, to_TO give_VVI a_AT1 (_( 3_MC 4_MC )_) submatrix_VV0 in_II the_AT top_JJ left_JJ corner_NN1 of_IO A._NP1 Again_RT ,_, since_CS &lsqb;_( formula_NN1 &rsqb;_) it_PPH1 makes_VVZ no_AT different_JJ if_CS we_PPIS2 divide_VV0 the_AT range_NN1 ,_, s_ZZ1 ,_, of_IO summation_NN1 into_II subranges_NN2 ;_; each_DD1 subrange_NN1 can_VM be_VBI summed_VVN ,_, and_CC these_DD2 sums_NN2 added_VVD to_TO give_VVI the_AT total_NN1 ._. 
This_DD1 is_VBZ equivalent_JJ to_II vertical_JJ partitioning_NN1 in_II B_ZZ1 with_IW conformable_JJ partitioning_NN1 in_II C._NP1 An_AT1 example_NN1 will_VM make_VVI this_DD1 clear_JJ ._. 
The_AT product_NN1 BC_NP1 =_FO A_ZZ1 could_VM be_VBI &lsqb;_( formula_NN1 &rsqb;_) ,_, where_CS B_ZZ1 is_VBZ (_( 4_MC 5_MC )_) ,_, C_ZZ1 is_VBZ (_( 5_MC 6_MC )_) and_CC A_ZZ1 is_VBZ (_( 4_MC 6_MC )_) ._. 
Here_RL B_ZZ1 has_VHZ been_VBN partitioned_VVN between_II the_AT second_MD and_CC third_MD rows_NN2 to_TO give_VVI ,_, at_II the_AT top_NN1 ,_, a_AT1 (_( 2_MC 5_MC )_) submatrix_NN1 (_( ignore_VV0 the_AT vertical_JJ partition_NN1 for_IF the_AT moment_NN1 )_) ;_; C_NP1 between_II the_AT fourth_MD and_CC fifth_MD columns_NN2 to_TO give_VVI ,_, on_II the_AT left_JJ ,_, a_AT1 (_( 5_MC 4_MC )_) submatrix_VV0 ._. 
Multiplied_VVN together_RL ,_, these_DD2 give_VV0 the_AT (_( 2_MC 4_MC )_) submatrix_VV0 at_II the_AT top_NN1 left_VVN in_II A._NP1 The_AT vertical_JJ partition_NN1 in_II B_ZZ1 ,_, after_CS the_AT third_MD column_NN1 ,_, and_CC the_AT corresponding_JJ partition_NN1 in_II C_NP1 ,_, which_DDQ must_VM be_VBI after_II the_AT third_MD row_NN1 ,_, merely_RR divide_VV0 each_DD1 summation_NN1 into_II two_MC parts_NN2 ._. 
Symbolically_RR ,_, we_PPIS2 may_VM write_VVI the_AT operation_NN1 as_CSA &lsqb;_( formula_NN1 &rsqb;_) ,_, where_CS the_AT elements_NN2 &lsqb;_( formula_NN1 &rsqb;_) etc_RA are_VBR now_RT the_AT above_JJ submatrices_NN2 ._. 
It_PPH1 is_VBZ important_JJ to_TO note_VVI the_AT conformability_NN1 ;_; thus_RR &lsqb;_( formula_NN1 &rsqb;_) is_VBZ (_( 2_MC 2_MC )_) (_( 2_MC 4_MC )_) =_FO (_( 2_MC 4_MC )_) also_RR ;_; and_CC &lsqb;_( formula_NN1 &rsqb;_) is_VBZ also_RR (_( 2_MC 4_MC )_) ._. 
It_PPH1 is_VBZ evident_JJ that_CST ,_, though_CS partitioning_NN1 is_VBZ largely_RR arbitrary_JJ ,_, conformability_NN1 both_RR in_II multiplication_NN1 and_CC in_RR21 addition_RR22 is_VBZ essential_JJ ._. 
Partitioning_NN1 is_VBZ often_RR of_IO great_JJ use_NN1 ,_, both_RR in_II analytical_JJ manipulation_NN1 of_IO matrices_NN2 and_CC in_II numerical_JJ cases_NN2 ,_, especially_RR when_CS submatrices_NN2 can_VM be_VBI chosen_VVN having_VHG simple_JJ forms_NN2 ,_, such_II21 as_II22 a_AT1 unit_NN1 or_CC a_AT1 null_JJ submatrix._NNU 1.5_MC SINGULAR_JJ MATRICES_NN2 ,_, DEGENERACY_NN1 AND_CC RANK_NN1 In_II a_AT1 square_JJ matrix_NN1 A_ZZ1 of_IO order_NN1 n_ZZ1 the_AT array_NN1 of_IO elements_NN2 has_VHZ a_AT1 determinant_NN1 &lsqb;_( formula_NN1 &rsqb;_) which_DDQ is_VBZ in_RR21 general_RR22 nonzero_NN1 :_: this_DD1 implies_VVZ that_CST the_AT columns_NN2 (_( and_CC the_AT rows_NN2 )_) are_VBR linearly_RR independent_JJ ,_, so_CS21 that_CS22 there_EX is_VBZ no_AT nonzero_NN1 column_NN1 vector_NN1 x_ZZ1 such_CS21 that_CS22 Ax_NP1 vanishes_VVZ ._. 
If_CS ,_, however_RR ,_, a_AT1 vector_NN1 x_ZZ1 does_VDZ exist_VVI for_IF which_DDQ Ax_NP1 =_FO 0_MC ,_, this_DD1 clearly_RR means_VVZ that_CST any_DD column_NN1 of_IO A_ZZ1 can_VM be_VBI expressed_VVN as_II a_AT1 linear_JJ sum_NN1 of_IO the_AT remaining_JJ columns_NN2 ;_; by_II the_AT rules_NN2 for_IF evaluation_NN1 of_IO determinants_NN2 this_DD1 requires_VVZ &lsqb;_( formula_NN1 &rsqb;_) to_TO be_VBI zero_MC ._. 
If_CS two_MC different_JJ vectors_NN2 x_ZZ1 exist_VV0 for_IF which_DDQ Ax_NP1 =_FO 0_MC ,_, then_RT not_XX only_RR is_VBZ zero_MC ,_, but_CCB also_RR all_DB the_AT minor_JJ determinants_NN2 of_IO order_NN1 (_( n_ZZ1 1_MC1 )_) vanish_VV0 ._. 
When_RRQ &lsqb;_( formula_NN1 &rsqb;_) ,_, the_AT matrix_NN1 A_ZZ1 is_VBZ said_VVN to_TO be_VBI singular_JJ ._. 
If_CS only_RR one_MC1 vector_NN1 x_ZZ1 exists_VVZ for_IF which_DDQ Ax_NP1 =_FO 0_MC ,_, A_ZZ1 is_VBZ said_VVN to_TO be_VBI simply_RR degenerate_JJ ,_, or_CC to_TO have_VHI simple_JJ degeneracy_NN1 ;_; if_CS more_DAR than_CSN one_MC1 such_DA vector_NN1 x_ZZ1 exists_VVZ ,_, A_ZZ1 has_VHZ multiple_JJ degeneracy_NN1 ._. 
Complementary_JJ to_II the_AT definition_NN1 of_IO degeneracy_NN1 is_VBZ that_DD1 of_IO rank_NN1 :_: for_IF a_AT1 square_JJ matrix_NN1 ,_, the_AT sum_NN1 of_IO the_AT degeneracy_NN1 and_CC rank_NN1 is_VBZ the_AT order_NN1 of_IO the_AT matrix_NN1 ;_; thus_RR &lsqb;_( formula_NN1 &rsqb;_) The_AT rank_NN1 is_VBZ in_II fact_NN1 the_AT order_NN1 of_IO the_AT largest_JJT nonvanishing_JJ minor_NN1 of_IO &lsqb;_( formula_NN1 &rsqb;_) ,_, so_CS that_DD1 rank_NN1 0_MC implies_VVZ a_AT1 null_JJ matrix_NN1 ._. 
We_PPIS2 shall_VM be_VBI much_RR concerned_JJ in_II this_DD1 text_NN1 with_IW matrices_NN2 which_DDQ are_VBR simply_RR degenerate_JJ (_( degeneracy_NN1 1_MC1 )_) and_CC with_IW matrices_NN2 in_II which_DDQ all_DB columns_NN2 (_( and_CC all_DB rows_NN2 )_) are_VBR proportional_JJ to_II each_PPX221 other_PPX222 ;_; in_II the_AT latter_DA case_NN1 all_DB second-order_JJ minors_NN2 vanish_VV0 ,_, so_CS21 that_CS22 the_AT rank_NN1 is_VBZ 1_MC1 ._. 
In_RR21 general_RR22 ,_, for_IF a_AT1 rectangular_JJ matrix_NN1 ,_, the_AT rank_NN1 is_VBZ the_AT number_NN1 of_IO linearly_RR independent_JJ rows_NN2 (_( and_CC columns_NN2 )_) ._. 
1.6_MC FACTORISATION_NN1 OF_IO MATRICES_NN2 All_DB matrices_NN2 can_VM be_VBI factorised_VVN ,_, usually_RR in_II a_AT1 variety_NN1 of_IO ways_NN2 ._. 
In_II the_AT simplest_JJT (_( trivial_JJ )_) case_NN1 we_PPIS2 may_VM write_VVI A_ZZ1 =_FO AI_NN1 =_FO IA_NP1 where_RRQ I_ZZ1 is_VBZ the_AT conformable_JJ unit_NN1 matrix_NN1 ._. 
Even_RR I_PPIS1 itself_PPX1 can_VM be_VBI factorised_VVN in_II a_AT1 variety_NN1 of_IO ways_NN2 ;_; as_CSA one_MC1 example_NN1 ,_, if_CS J_ZZ1 is_VBZ the_AT square_JJ matrix_NN1 which_DDQ has_VHZ units_NN2 in_II its_APPGE secondary_JJ diagonal_JJ and_CC zeros_MC2 elsewhere_RL ,_, which_DDQ we_PPIS2 shall_VM call_VVI the_AT reversing_NN1 matrix_NN1 ,_, then_RT &lsqb;_( formula_NN1 &rsqb;_) ;_; e.g._REX for_IF order_NN1 3_MC &lsqb;_( formula_NN1 &rsqb;_) so_CS21 that_CS22 J_ZZ1 is_VBZ a_AT1 double_JJ factor_NN1 of_IO I._NP1 The_AT ability_NN1 to_TO factorise_VVI a_AT1 matrix_NN1 appropriately_RR can_VM often_RR be_VBI of_IO great_JJ assistance_NN1 in_II solving_VVG practical_JJ problems_NN2 ._. 
Singular_JJ matrices_NN2 can_VM be_VBI factorised_VVN as_II the_AT product_NN1 of_IO two_MC rectangular_JJ matrices_NN2 again_RT ,_, in_II more_DAR ways_NN2 than_CSN one_PN1 the_AT orders_NN2 of_IO which_DDQ are_VBR determined_VVN by_II the_AT rank_NN1 of_IO the_AT matrix_NN1 ._. 
This_DD1 may_VM be_VBI illustrated_VVN by_II a_AT1 simple_JJ numerical_JJ example_NN1 ._. 
Let_VV0 &lsqb;_( formula_NN1 &rsqb;_) Inspection_NN1 shows_VVZ the_AT third_MD column_NN1 to_TO be_VBI the_AT sum_NN1 of_IO the_AT first_MD two_MC ._. 
Hence_RR if_CS we_PPIS2 partition_VV0 the_AT matrix_NN1 appropriately_RR ,_, &lsqb;_( formula_NN1 &rsqb;_) where_RRQ the_AT submatrix_NN1 products_NN2 are_VBR written_VVN separately_RR ._. 
If_CS now_RT we_PPIS2 postmultiply_RR the_AT (_( 3_MC 2_MC )_) submatrix_NN1 of_IO A_ZZ1 by_II I_ZZ1 ,_, we_PPIS2 can_VM add_VVI the_AT last_MD result_NN1 to_II it_PPH1 to_TO recover_VVI A_ZZ1 as_II the_AT matrix_NN1 product_NN1 &lsqb;_( formula_NN1 &rsqb;_) Finally_RR ,_, a_AT1 matrix_NN1 of_IO rank_NN1 1_MC1 can_VM be_VBI expressed_VVN as_II the_AT product_NN1 of_IO a_AT1 column_NN1 and_CC a_AT1 row_NN1 in_II that_DD1 order_NN1 for_IF such_DA a_AT1 matrix_NN1 has_VHZ effectively_RR only_RR one_MC1 independent_JJ column_NN1 ,_, all_DB the_AT other_JJ columns_NN2 being_VBG proportional_JJ to_II it_PPH1 ;_; similarly_RR for_IF the_AT rows_NN2 ._. 
For_REX21 example_REX22 &lsqb;_( formula_NN1 &rsqb;_) In_RR21 general_RR22 ,_, any_DD singular_JJ matrix_NN1 of_IO order_NN1 n_ZZ1 and_CC rank_NN1 r_ZZ1 is_VBZ expressible_JJ in_II an_AT1 infinite_JJ number_NN1 of_IO ways_NN2 as_II the_AT product_NN1 of_IO two_MC matrices_NN2 of_IO order_NN1 (_( n_ZZ1 r_ZZ1 )_) and_CC (_( r_ZZ1 n_ZZ1 )_) ,_, since_CS r_ZZ1 is_VBZ the_AT number_NN1 of_IO independent_JJ columns_NN2 (_( and_CC rows_NN2 )_) ._. 
1.7_MC ADJUGATE_NP1 AND_CC RECIPROCAL_JJ MATRICES_NN2 If_CS &lsqb;_( formula_NN1 &rsqb;_) is_VBZ a_AT1 square_JJ matrix_NN1 of_IO order_NN1 n_ZZ1 ,_, and_CC if_CS ,_, in_II the_AT determinant_NN1 &lsqb;_( formula_NN1 &rsqb;_) ,_, the_AT element_NN1 &lsqb;_( formula_NN1 &rsqb;_) has_VHZ the_AT cofactor_NN1 &lsqb;_( formula_NN1 &rsqb;_) (_( that_REX21 is_REX22 ,_, the_AT value_NN1 of_IO the_AT determinant_NN1 obtained_VVN from_II &lsqb;_( formula_NN1 &rsqb;_) by_II deleting_VVG the_AT ith_MD row_NN1 and_CC jth_MD column_NN1 ,_, leaving_VVG a_AT1 minor_NN1 of_IO order_NN1 (_( n_ZZ1 1_MC1 )_) which_DDQ is_VBZ then_RT multiplied_VVN by_II &lsqb;_( formula_NN1 &rsqb;_) ,_, then_RT by_II the_AT rules_NN2 for_IF evaluation_NN1 of_IO determinants_NN2 &lsqb;_( formula_NN1 &rsqb;_) while_CS &lsqb;_( formula_NN1 &rsqb;_) since_CS the_AT summation_NN1 in_II the_AT latter_DA result_NN1 gives_VVZ the_AT value_NN1 of_IO a_AT1 determinant_NN1 having_VHG two_MC equal_JJ rows_NN2 ._. 
It_PPH1 follows_VVZ that_CST if_CS the_AT cofactors_NN2 &lsqb;_( formula_NN1 &rsqb;_) are_VBR arranged_VVN as_II a_AT1 matrix_NN1 &lsqb;_( formula_NN1 &rsqb;_) of_IO order_NN1 n_ZZ1 ,_, then_RT &lsqb;_( formula_NN1 &rsqb;_) where_RRQ ,_, to_TO conform_VVI with_IW matrix_NN1 multiplication_NN1 rules_NN2 ,_, we_PPIS2 must_VM use_VVI the_AT transpose_VV0 of_IO &lsqb;_( formula_NN1 &rsqb;_) to_II postmultiply_RR A._NNU If_CS instead_II21 of_II22 a_AT1 row_NN1 summation_NN1 as_CSA in_II (_( 1_MC1 )_) we_PPIS2 use_VV0 column_NN1 summation_NN1 ,_, we_PPIS2 find_VV0 also_RR &lsqb;_( formula_NN1 &rsqb;_) so_CS21 that_CS22 A_ZZ1 and_CC &lsqb;_( formula_NN1 &rsqb;_) are_VBR permutable_JJ ._. 
The_AT matrix_NN1 &lsqb;_( formula_NN1 &rsqb;_) is_VBZ called_VVN the_AT adjugate_NN1 of_IO A._NNU If_CS A_ZZ1 is_VBZ singular_JJ ,_, so_CS21 that_CS22 &lsqb;_( formula_NN1 &rsqb;_) vanishes_VVZ ,_, then_RT &lsqb;_( formula_NN1 &rsqb;_) In_II this_DD1 case_NN1 ,_, if_CS A_ZZ1 is_VBZ simply_RR degenerate_JJ ,_, &lsqb;_( formula_NN1 &rsqb;_) is_VBZ of_IO rank_NN1 1_MC1 ;_; but_CCB if_CS A_ZZ1 is_VBZ multiply_RR degenerate_JJ ,_, so_CS21 that_CS22 all_DB the_AT cofactors_NN2 of_IO Aij_NP1 vanish_VV0 for_IF all_DB i_ZZ1 ,_, j_ZZ1 ,_, then_RT &lsqb;_( formula_NN1 &rsqb;_) is_VBZ null_JJ ._. 
These_DD2 results_NN2 are_VBR examples_NN2 of_IO Sylvester_NP1 's_GE law_NN1 of_IO degeneracy_NN1 (_( see_VV0 Theorem_NN1 XII_MC of_IO 1.22_MC )_) ,_, viz._REX the_AT degeneracy_NN1 of_IO the_AT product_NN1 of_IO two_MC matrices_NN2 is_VBZ at_RR21 least_RR22 as_RG great_JJ as_CSA the_AT degeneracy_NN1 of_IO either_DD1 factor_NN1 ,_, and_CC at_RR21 most_RR22 as_RG great_JJ as_CSA the_AT sum_NN1 of_IO the_AT degeneracies_NN2 of_IO the_AT factors_NN2 ._. 
When_RRQ &lsqb;_( formula_NN1 &rsqb;_) ,_, we_PPIS2 may_VM divide_VVI &lsqb;_( formula_NN1 &rsqb;_) by_II &lsqb;_( formula_NN1 &rsqb;_) ,_, calling_VVG the_AT result_NN1 R._NP1 then_RT AR_UH =_FO I_ZZ1 ,_, and_CC R_ZZ1 can_VM be_VBI described_VVN as_II the_AT reciprocal_JJ or_CC the_AT inverse_NN1 of_IO A._NP1 We_PPIS2 have_VH0 already_RR seen_VVN that_CST it_PPH1 permutes_VVZ with_IW A_ZZ1 ;_; however_RR ,_, this_DD1 is_VBZ readily_RR proved_VVN alternatively_RR ._. 
Let_VV0 &lsqb;_( formula_NN1 &rsqb;_) from_II the_AT latter_DA ,_, postmultiplying_VVG by_II R1_FO ,_, we_PPIS2 have_VH0 R2AR_FO =_FO R1_FO ,_, and_CC using_VVG the_AT former_DA ,_, R2_FO =_FO R1_FO ._. 
Moreover_RR ,_, the_AT reciprocal_JJ is_VBZ unique_JJ ;_; for_IF if_CS AR1_FO =_FO I_ZZ1 and_CC AR2_FO =_FO I_ZZ1 ,_, then_RT A_ZZ1 (_( R1_FO R2_FO )_) =_FO 0_MC ,_, and_CC on_II premultiplication_NN1 by_II R1_FO we_PPIS2 have_VH0 R1_FO =_FO R2_FO ._. 
In_II conformity_NN1 with_IW usual_JJ algebraic_JJ notation_NN1 ,_, the_AT reciprocal_JJ of_IO A_ZZ1 is_VBZ written_VVN &lsqb;_( formula_NN1 &rsqb;_) so_CS21 that_CS22 a_AT1 separate_JJ symbol_NN1 is_VBZ unnecessary_JJ ._. 
Thus_RR &lsqb;_( formula_NN1 &rsqb;_) expresses_VVZ the_AT essential_JJ property_NN1 of_IO the_AT reciprocal_JJ of_IO A._NP1 Just_RR as_CSA with_IW transposition_NN1 ,_, if_CS a_AT1 matrix_NN1 product_NN1 is_VBZ inverted_JJ the_AT order_NN1 of_IO the_AT factors_NN2 must_VM be_VBI reversed_VVN ._. 
Let_VV0 A_ZZ1 =_FO BCD_NP1 ,_, where_CS the_AT matrices_NN2 are_VBR all_DB square_JJ and_CC non-singular_NN1 ._. 
By_II premultiplication_NN1 or_CC postmultiplication_NN1 as_CSA required_VVN ,_, we_PPIS2 have_VH0 successively_RR &lsqb;_( formula_NN1 &rsqb;_) 1.8_MC POWERS_NN2 OF_IO MATRICES_NN2 ,_, POLYNOMIALS_NN2 AND_CC SERIES_NN We_PPIS2 have_VH0 just_RR established_VVN the_AT existence_NN1 of_IO a_AT1 negative_JJ power_NN1 of_IO a_AT1 square_JJ non-singular_JJ matrix_NN1 ._. 
In_RR21 general_RR22 ,_, any_DD square_JJ matrix_NN1 can_VM be_VBI raised_VVN to_II any_DD positive_JJ integral_JJ power_NN1 by_II continued_JJ multiplication_NN1 ,_, in_II any_DD order_NN1 ,_, by_II itself_PPX1 ._. 
Any_DD such_DA powers_NN2 are_VBR evidently_RR permutable_JJ ;_; for_REX21 example_REX22 &lsqb;_( formula_NN1 &rsqb;_) Similarly_RR ,_, any_DD power_NN1 of_IO a_AT1 matrix_NN1 may_VM itself_PPX1 be_VBI raised_VVN to_II a_AT1 power_NN1 ;_; for_REX21 example_REX22 &lsqb;_( formula_NN1 &rsqb;_) while_CS if_CS A_ZZ1 is_VBZ non-singular_JJ &lsqb;_( formula_NN1 &rsqb;_) Evidently_RR ,_, the_AT index_NN1 laws_NN2 of_IO ordinary_JJ algebra_NN1 apply_VV0 ._. 
In_RR21 particular_RR22 ,_, &lsqb;_( formula_NN1 &rsqb;_) where_RRQ A_ZZ1 is_VBZ non-singular_JJ ;_; in_II fact_NN1 &lsqb;_( formula_NN1 &rsqb;_) applies_VVZ for_IF all_DB square_JJ matrices_NN2 ._. 
Just_RR as_CSA we_PPIS2 can_VM have_VHI polynomials_NN2 involving_VVG powers_NN2 of_IO a_AT1 scalar_JJ quantity_NN1 gl_NNU ,_, e.g._REX &lsqb;_( formula_NN1 &rsqb;_) so_CS we_PPIS2 can_VM have_VHI polynomials_NN2 of_IO a_AT1 square_JJ matrix_NN1 A_ZZ1 :_: &lsqb;_( formula_NN1 &rsqb;_) where_RRQ the_AT &lsqb;_( formula_NN1 &rsqb;_) are_VBR scalar_JJ multipliers_NN2 ._. 
Polynomials_NN2 of_IO matrices_NN2 are_VBR of_IO great_JJ importance_NN1 in_II a_AT1 variety_NN1 of_IO ways_NN2 ._. 
For_REX21 example_REX22 ,_, suppose_VV0 we_PPIS2 can_VM find_VVI a_AT1 set_NN1 of_IO multipliers_NN2 pi_NN1 such_CS21 that_CS22 P(A)_NN1 vanishes_VVZ ._. 
Then_RT ,_, if_CS &lsqb;_( formula_NN1 &rsqb;_) exists_VVZ ,_, we_PPIS2 may_VM premultiply_RR or_CC postmultiply_JJ P(A)_NN1 by_II &lsqb;_( formula_NN1 &rsqb;_) to_TO obtain_VVI &lsqb;_( formula_NN1 &rsqb;_) as_II a_AT1 convenient_JJ formula_NN1 for_IF computing_NN1 &lsqb;_( formula_NN1 &rsqb;_) ._. 
We_PPIS2 can_VM also_RR define_VVI matrix_NN1 series_NN ;_; for_REX21 example_REX22 the_AT parallel_JJ series_NN &lsqb;_( formula_NN1 &rsqb;_) exp_JJ A_ZZ1 converges_VVZ in_RP exactly_RR the_AT same_DA way_NN1 as_CSA exp_JJ gl_NNU ._. 
In_II parallel_NN1 with_IW the_AT scalar_JJ series_NN ,_, exp_JJ (_( -_- A_ZZ1 )_) is_VBZ the_AT reciprocal_JJ of_IO exp_JJ A._NNU If_CS ,_, as_CSA in_II (_( 2_MC )_) and_CC (_( 3_MC )_) ,_, we_PPIS2 have_VH0 a_AT1 polynomial_NN1 or_CC series_NN in_II a_AT1 scalar_JJ quantity_NN1 gl_NNU and_CC the_AT corresponding_JJ function_NN1 in_II the_AT square_JJ matrix_NN1 A_ZZ1 ,_, we_PPIS2 may_VM multiply_VVI the_AT scalar_JJ function_NN1 by_II I_MC1 to_TO obtain_VVI the_AT two_MC matrix_NN1 functions_NN2 &lsqb;_( formula_NN1 &rsqb;_) Subtraction_NN1 term_NN1 by_II term_NN1 then_RT gives_VVZ &lsqb;_( formula_NN1 &rsqb;_) where_RRQ Q_ZZ1 is_VBZ a_AT1 function_NN1 we_PPIS2 need_VM not_XX evaluate_VVI here_RL ._. 
The_AT matrix_NN1 &lsqb;_( formula_NN1 &rsqb;_) is_VBZ of_IO very_RG great_JJ importance_NN1 ,_, not_XX only_RR in_II studies_NN2 of_IO the_AT intrinsic_JJ properties_NN2 of_IO A_ZZ1 ,_, but_CCB also_RR in_II dynamical_JJ and_CC other_JJ applications_NN2 it_PPH1 is_VBZ called_VVN the_AT characteristic_JJ matrix_NN1 of_IO A._NP1 1.9_MC SPECIAL_JJ TYPES_NN2 OF_IO MATRIX_NN1 We_PPIS2 have_VH0 already_RR defined_VVN symmetric_JJ ,_, skew-symmetric_JJ ,_, diagonal_JJ ,_, unit_NN1 and_CC null_JJ matrices_NN2 ._. 
We_PPIS2 now_RT summarise_VV0 and_CC extend_VV0 these_DD2 definitions._NNU (_( 1_MC1 )_) Symmetric_JJ matrix_NN1 characterised_VVN by_II A_ZZ1 =_FO AT._NP1 (_( 2_MC )_) Skew-symmetric_JJ matrix_NN1 characterised_VVN by_II A_ZZ1 =_FO AT._NP1 (_( 3_MC )_) Diagonal_JJ matrix_NN1 characterised_VVN by_II &lsqb;_( formula_NN1 &rsqb;_) ._. 
We_PPIS2 now_RT establish_VV0 an_AT1 important_JJ property_NN1 of_IO diagonal_JJ matrices_NN2 ._. 
Let_VV0 C_ZZ1 be_VBI any_DD square_JJ matrix_NN1 and_CC D_ZZ1 a_AT1 conformable_JJ diagonal_JJ matrix_NN1 with_IW diagonal_JJ elements_NN2 &lsqb;_( formula_NN1 &rsqb;_) ._. 
Then_RT postmultiplication_NN1 of_IO C_ZZ1 by_II D_ZZ1 multiplies_VVZ the_AT columns_NN2 of_IO C_ZZ1 in_II order_NN1 by_II &lsqb;_( formula_NN1 &rsqb;_) ,_, etc._RA ,_, while_CS premultiplication_NN1 of_IO C_ZZ1 by_II D_ZZ1 multiplies_VVZ the_AT rows_NN2 of_IO C_ZZ1 in_II order_NN1 by_II &lsqb;_( formula_NN1 &rsqb;_) ,_, etc_RA ._. 
Thus_RR ,_, the_AT typical_JJ element_NN1 of_IO CD_NN1 is_VBZ &lsqb;_( formula_NN1 &rsqb;_) while_CS that_DD1 of_IO DC_NP1 is_VBZ &lsqb;_( formula_NN1 &rsqb;_) ._. 
It_PPH1 follows_VVZ that_CST if_CS C_ZZ1 and_CC D_ZZ1 permute_VV0 ,_, so_CS21 that_CS22 DC_NP1 =_FO CD_NN1 ,_, then_RT &lsqb;_( formula_NN1 &rsqb;_) if_CS &lsqb;_( formula_NN1 &rsqb;_) ._. 
For_IF a_AT1 diagonal_JJ matrix_NN1 D_NP1 having_VHG all_RR its_APPGE diagonal_JJ elements_NN2 different_JJ ,_, it_PPH1 follows_VVZ that_CST C_ZZ1 must_VM then_RT also_RR be_VBI diagonal_JJ ._. 
If_CS D_ZZ1 has_VHZ some_DD equal_JJ diagonal_JJ elements_NN2 ,_, then_RT C_NP1 can_VM possess_VVI nonzero_NN1 off-diagonal_JJ elements_NN2 in_II a_AT1 &quot;_" diagonal_JJ submatrix_NN1 &quot;_" corresponding_VVG to_II the_AT equal_JJ elements_NN2 of_IO D._NP1 For_REX21 example_REX22 ,_, if_CS &lsqb;_( formula_NN1 &rsqb;_) which_DDQ we_PPIS2 write_VV0 as_CSA D_ZZ1 =_FO Diag(1,2,3)_FO ,_, and_CC &lsqb;_( formula_NN1 &rsqb;_) then_RT if_CS C_ZZ1 and_CC D_ZZ1 are_VBR to_TO permute_VVI we_PPIS2 must_VM have_VHI &lsqb;_( formula_NN1 &rsqb;_) so_CS21 that_CS22 d_ZZ1 =_FO e_ZZ1 =_FO f_ZZ1 =_FO g_ZZ1 =_FO h_ZZ1 =_FO k_ZZ1 =_FO 0_MC ,_, leaving_VVG C_ZZ1 =_FO Diag_NP1 (_( a_AT1 ,_, b_ZZ1 ,_, c_ZZ1 )_) ._. 
However_RR ,_, if_CS D_ZZ1 =_FO Diag(1,2,3)_FO we_PPIS2 have_VH0 that_DD1 &lsqb;_( formula_NN1 &rsqb;_) so_CS21 that_CS22 e_ZZ1 =_FO g_ZZ1 =_FO h_ZZ1 =_FO k_ZZ1 =_FO 0_MC leaving_NN1 &lsqb;_( formula_NN1 &rsqb;_) which_DDQ we_PPIS2 may_VM write_VVI as_II Block_NN1 Diag_NN1 &lsqb;_( formula_NN1 &rsqb;_) where_RRQ &lsqb;_( formula_NN1 &rsqb;_) Finally_RR ,_, we_PPIS2 note_VV0 that_CST if_CS &lsqb;_( formula_NN1 &rsqb;_) for_IF all_DB i_ZZ1 ,_, then_RT &lsqb;_( formula_NN1 &rsqb;_) ._. 
This_DD1 follows_VVZ from_II the_AT identity_NN1 &lsqb;_( formula_NN1 &rsqb;_) ._. 
(_( 4_MC )_) Unit_NN1 matrix_NN1 this_DD1 is_VBZ the_AT diagonal_JJ matrix_NN1 with_IW units_NN2 in_II the_AT diagonal_JJ positions_NN2 ;_; it_PPH1 is_VBZ written_VVN I._NP1 (_( 5_MC )_) Scalar_JJ matrix_NN1 this_DD1 is_VBZ the_AT diagonal_JJ matrix_NN1 with_IW the_AT same_DA scalar_JJ quantity_NN1 ,_, say_VV0 &lsqb;_( formula_NN1 &rsqb;_) ,_, in_II the_AT diagonal_JJ positions_NN2 ._. 
Evidently_RR such_DA a_AT1 matrix_NN1 may_VM be_VBI written_VVN &lsqb;_( formula_NN1 &rsqb;_) ;_; multiplication_NN1 by_II a_AT1 scalar_JJ matrix_NN1 therefore_RR implies_VVZ multiplication_NN1 by_II a_AT1 scalar_JJ quantity._NNU (_( 6_MC )_) Reversing_VVG matrix_NN1 this_DD1 is_VBZ the_AT square_JJ matrix_NN1 having_VHG units_NN2 in_II its_APPGE secondary_JJ diagonal_JJ and_CC zeros_MC2 elsewhere_RL ;_; it_PPH1 is_VBZ written_VVN J._NP1 Postmultiplication_NN1 of_IO any_DD matrix_NN1 C_ZZ1 by_II the_AT conformable_JJ J_ZZ1 matrix_NN1 reverses_VVZ the_AT order_NN1 of_IO the_AT columns_NN2 of_IO C_ZZ1 while_CS premultiplication_NN1 by_II J_ZZ1 reverses_VVZ the_AT order_NN1 of_IO the_AT rows._NNU (_( 7_MC )_) Null_JJ matrix_NN1 this_DD1 has_VHZ zero_MC elements_NN2 throughout._NNU (_( 8_MC )_) Triangular_JJ matrices_NN2 these_DD2 are_VBR of_IO two_MC types_NN2 :_: lower_JJR triangular_JJ ,_, L_ZZ1 ,_, and_CC upper_JJ triangular_JJ ,_, U._NP1 L_ZZ1 is_VBZ characterised_VVN by_II a_AT1 typical_JJ element_NN1 having_VHG the_AT property_NN1 &lsqb;_( formula_NN1 &rsqb;_) while_CS U_ZZ1 is_VBZ characterised_VVN by_II &lsqb;_( formula_NN1 &rsqb;_) ._. 
Thus_RR ,_, for_REX21 example_REX22 &lsqb;_( formula_NN1 &rsqb;_) are_VBR ,_, respectively_RR ,_, lower_JJR and_CC upper_JJ triangular_JJ matrices_NN2 ._. 
An_AT1 obvious_JJ and_CC useful_JJ property_NN1 of_IO triangular_JJ matrices_NN2 is_VBZ that_CST their_APPGE determinants_NN2 are_VBR simply_RR the_AT products_NN2 of_IO their_APPGE diagonal_JJ terms_NN2 ._. 
Thus_RR ,_, in_II the_AT above_JJ example_NN1 ,_, &lsqb;_( formula_NN1 &rsqb;_) Another_DD1 useful_JJ property_NN1 of_IO triangular_JJ matrices_NN2 is_VBZ the_AT ease_NN1 with_IW which_DDQ their_APPGE reciprocals_NN2 may_VM be_VBI calculated_VVN ._. 
For_REX21 example_REX22 ,_, if_CS R_ZZ1 =_FO (_( Rij_NP1 )_) is_VBZ the_AT reciprocal_JJ of_IO a_AT1 lower_JJR triangular_JJ matrix_NN1 L_ZZ1 then_RT &lsqb;_( formula_NN1 &rsqb;_) where_RRQ a_AT1 ,_, b_ZZ1 ,_, c_ZZ1 ,_, ..._... must_VM be_VBI nonzero_NN1 ;_; otherwise_RR &lsqb;_( formula_NN1 &rsqb;_) and_CC R_ZZ1 does_VDZ not_XX exist_VVI ._. 
The_AT first_MD row_NN1 of_IO the_AT product_NN1 gives_VVZ &lsqb;_( formula_NN1 &rsqb;_) and_CC in_II31 view_II32 of_II33 this_DD1 ,_, the_AT second_MD row_NN1 ,_, excluding_VVG its_APPGE first_MD element_NN1 ,_, gives_VVZ &lsqb;_( formula_NN1 &rsqb;_) and_RR31 so_RR32 on_RR33 ._. 
Accordingly_RR ,_, R_ZZ1 is_VBZ also_RR lower_JJR triangular_JJ ,_, and_CC its_APPGE diagonal_JJ elements_NN2 are_VBR &lsqb;_( formula_NN1 &rsqb;_) ,_, as_CSA is_VBZ otherwise_RR obvious_JJ from_II consideration_NN1 of_IO the_AT cofactors_NN2 of_IO a_AT1 ,_, b_ZZ1 ,_, c_ZZ1 ,_, ..._... in_II L._NP1 Hence_RR a_AT1 computer_NN1 program_NN1 to_TO find_VVI R_ZZ1 needs_VVZ to_TO calculate_VVI only_RR the_AT n_ZZ1 (_( n_ZZ1 2_MC )_) /2_MF elements_NN2 below_II the_AT diagonal_JJ ;_; and_CC these_DD2 are_VBR found_VVN progressively_RR ._. 
For_REX21 example_REX22 &lsqb;_( formula_NN1 &rsqb;_) gives_VVZ ,_, from_II the_AT first_MD column_NN1 ,_, &lsqb;_( formula_NN1 &rsqb;_) and_CC from_II the_AT second_MD column_NN1 ,_, &lsqb;_( formula_NN1 &rsqb;_) Transposition_NN1 of_IO the_AT above_JJ shows_NN2 that_CST the_AT reciprocal_JJ of_IO an_AT1 upper_JJ triangular_JJ matrix_NN1 U_ZZ1 is_VBZ itself_PPX1 upper_JJ triangular_JJ ,_, and_CC that_CST its_APPGE diagonal_JJ elements_NN2 are_VBR the_AT reciprocals_NN2 of_IO those_DD2 of_IO U._NP1 (_( 9_MC )_) Persymmetric_JJ matrix_NN1 this_DD1 is_VBZ a_AT1 square_JJ matrix_NN1 in_II which_DDQ the_AT elements_NN2 in_II any_DD line_NN1 parallel_NN1 to_II the_AT secondary_JJ diagonal_JJ are_VBR equal_JJ ._. 
Thus_RR ,_, if_CS A_ZZ1 is_VBZ persymmetric_JJ ,_, it_PPH1 is_VBZ characterised_VVN by_II the_AT typical_JJ element_NN1 &lsqb;_( formula_NN1 &rsqb;_) and_CC hence_RR has_VHZ ,_, in_RR21 general_RR22 ,_, 2n_FO 1_MC1 independent_JJ elements_NN2 ._. 
For_REX21 example_REX22 ,_, the_AT matrix_NN1 &lsqb;_( formula_NN1 &rsqb;_) is_VBZ persymmetric._NNU (_( 10_MC )_) Centrosymmetric_NP1 and_CC centroskew_VV0 matrices_NN2 a_AT1 matrix_NN1 which_DDQ is_VBZ symmetric_JJ about_II the_AT centre_NN1 point_NN1 of_IO its_APPGE array_NN1 is_VBZ said_VVN to_TO be_VBI centrosymmetric_JJ ;_; thus_RR ,_, if_CS A_ZZ1 is_VBZ centrosymmetric_JJ ,_, of_IO order_NN1 n_ZZ1 ,_, it_PPH1 is_VBZ characterised_VVN by_II &lsqb;_( formula_NN1 &rsqb;_) More_RGR simply_RR ,_, if_CS J_ZZ1 is_VBZ the_AT reversing_NN1 matrix_NN1 ,_, a_AT1 centrosymmetric_JJ matrix_NN1 has_VHZ the_AT property_NN1 A_ZZ1 =_FO JAJ_NP1 A_ZZ1 centroskew_VV0 matrix_NN1 is_VBZ characterised_VVN by_II the_AT equation_NN1 A_ZZ1 =_FO JAJ_NP1 ._. 
For_REX21 example_REX22 ,_, the_AT matrices_NN2 &lsqb;_( formula_NN1 &rsqb;_) are_VBR centrosymmetric_JJ ,_, while_CS the_AT matrices_NN2 &lsqb;_( formula_NN1 &rsqb;_) are_VBR centroskew_NN1 ._. 
Centrosymmetric_JJ matrices_NN2 and_CC centroskew_VV0 vectors_NN2 arise_VV0 in_II the_AT study_NN1 of_IO the_AT dynamics_NN of_IO mechanical_JJ systems_NN2 having_VHG physical_JJ symmetry_NN1 (_( e.g._REX a_AT1 suspension_NN1 bridge_NN1 )_) ._. 
(_( 11_MC )_) Orthogonal_JJ matrices_NN2 a_AT1 square_NN1 ,_, non-singular_JJ matrix_NN1 A_ZZ1 having_VHG the_AT property_NN1 &lsqb;_( formula_NN1 &rsqb;_) is_VBZ said_VVN to_TO be_VBI orthogonal_JJ ._. 
Orthogonal_JJ matrices_NN2 are_VBR of_IO very_RG great_JJ importance_NN1 in_II dynamics_NN ._. 
For_REX21 example_REX22 ,_, the_AT matrix_NN1 &lsqb;_( formula_NN1 &rsqb;_) is_VBZ orthogonal_JJ ,_, since_CS &lsqb;_( formula_NN1 &rsqb;_) It_PPH1 should_VM be_VBI noted_VVN that_CST the_AT determinant_NN1 of_IO the_AT above_JJ orthogonal_JJ matrix_NN1 is_VBZ unity_NN1 :_: in_RR21 general_RR22 an_AT1 orthogonal_JJ matrix_NN1 will_VM have_VHI &lsqb;_( formula_NN1 &rsqb;_) as_CSA its_APPGE determinant_NN1 since_CS &lsqb;_( formula_NN1 &rsqb;_) (_( 12_MC )_) Hermitian_JJ and_CC skew-Hermitian_JJ matrices_NN2 if_CS a_AT1 matrix_NN1 ,_, A_ZZ1 ,_, has_VHZ complex_JJ elements_NN2 ,_, it_PPH1 can_VM clearly_RR be_VBI written_VVN in_II the_AT form_NN1 &lsqb;_( formula_NN1 &rsqb;_) where_RRQ B_ZZ1 and_CC C_ZZ1 are_VBR real_JJ matrices_NN2 and_CC &lsqb;_( formula_NN1 &rsqb;_) is_VBZ the_AT conjugate_NN1 of_IO A._NNU If_CS B_ZZ1 is_VBZ symmetric_JJ and_CC C_ZZ1 is_VBZ skew-symmetric_JJ ,_, A_ZZ1 is_VBZ said_VVN to_TO be_VBI Hermitian_JJ ._. 
Evidently_RR in_II this_DD1 case_NN1 &lsqb;_( formula_NN1 &rsqb;_) If_CS B_ZZ1 is_VBZ skew-symmetric_JJ and_CC C_ZZ1 is_VBZ symmetric_JJ ,_, A_ZZ1 is_VBZ said_VVN to_TO be_VBI skew-Hermitian_JJ ._. 
Evidently_RR in_II this_DD1 case_NN1 &lsqb;_( formula_NN1 &rsqb;_) For_REX21 example_REX22 ,_, the_AT matrix_NN1 &lsqb;_( formula_NN1 &rsqb;_) is_VBZ Hermitian_JJ ,_, while_CS the_AT matrix_NN1 &lsqb;_( formula_NN1 &rsqb;_) is_VBZ skew-Hermitian_JJ ._. 
It_PPH1 should_VM be_VBI noted_VVN that_CST if_CS A_ZZ1 is_VBZ Hermitian_JJ ,_, then_RT iA_NNU is_VBZ a_AT1 skew-Hermitian._NP1 (_( 13_MC )_) The_AT isolating_JJ vector_NN1 ei_NNU this_DD1 vector_NN1 is_VBZ the_AT ith_MD column_NN1 of_IO the_AT unit_NN1 matrix_NN1 ;_; it_PPH1 has_VHZ a_AT1 unit_NN1 in_II the_AT ith_MD element_NN1 and_CC zeros_MC2 elsewhere_RL ._. 
The_AT operation_NN1 &lsqb;_( formula_NN1 &rsqb;_) isolates_VVZ the_AT ith_MD row_NN1 of_IO A_ZZ1 ,_, while_CS Aej_NP1 isolates_VVZ the_AT jth_MD column_NN1 ._. 
The_AT combined_JJ operation_NN1 &lsqb;_( formula_NN1 &rsqb;_) gives_VVZ the_AT isolated_JJ element_NN1 of_IO &lsqb;_( formula_NN1 &rsqb;_) (_( 14_MC )_) The_AT summing_JJ vector_NN1 &lsqb;_( formula_NN1 &rsqb;_) this_DD1 is_VBZ the_AT vector_NN1 of_IO which_DDQ all_DB elements_NN2 are_VBR units_NN2 ._. 
Evidently_RR &lsqb;_( formula_NN1 &rsqb;_) gives_VVZ a_AT1 row_NN1 which_DDQ is_VBZ the_AT sum_NN1 of_IO all_DB the_AT rows_NN2 of_IO A_ZZ1 and_CC Ags_NP1 sums_VVZ all_DB the_AT columns_NN2 of_IO A._NP1 Its_APPGE principal_JJ use_NN1 is_VBZ in_II the_AT checking_NN1 of_IO numerical_JJ operations_NN2 ._. 
The_AT above_JJ list_NN1 is_VBZ by_RR31 no_RR32 means_RR33 exhaustive_JJ ,_, but_CCB it_PPH1 covers_VVZ most_DAT of_IO the_AT matrix_NN1 types_NN2 which_DDQ arise_VV0 in_II the_AT present_JJ text._NNU 1.10_MC VARIABLE_JJ MATRICES_NN2 ,_, LINEAR_JJ SUBSTITUTIONS_NN2 AND_CC ALGEBRAIC_JJ EQUATIONS_NN2 So_RG far_RR ,_, we_PPIS2 have_VH0 discussed_VVN the_AT properties_NN2 of_IO matrices_NN2 ,_, and_CC the_AT algebra_NN1 of_IO matrices_NN2 ,_, in_II general_JJ terms_NN2 ,_, and_CC without_IW reference_NN1 to_II the_AT nature_NN1 of_IO the_AT elements_NN2 :_: where_RRQ illustrations_NN2 have_VH0 been_VBN given_VVN we_PPIS2 have_VH0 used_VVN arithmetical_JJ numbers_NN2 for_IF elements_NN2 ._. 
But_CCB the_AT elements_NN2 can_VM of_RR21 course_RR22 be_VBI variables_NN2 ._. 
They_PPHS2 may_VM ,_, for_REX21 example_REX22 ,_, all_DB be_VBI functions_NN2 of_IO time_NNT1 ,_, so_CS21 that_CS22 we_PPIS2 must_VM write_VVI A_ZZ1 =_FO A(t)_II ;_; they_PPHS2 may_VM be_VBI functions_NN2 of_IO a_AT1 parameter_NN1 ,_, typified_VVN by_II &lsqb;_( formula_NN1 &rsqb;_) when_RRQ the_AT elements_NN2 are_VBR rational_JJ integral_JJ functions_NN2 of_IO gl_NNU ,_, the_AT matrix_NN1 is_VBZ called_VVN a_AT1 lambda-matrix_NN1 ;_; or_CC the_AT elements_NN2 may_VM themselves_PPX2 (_( especially_RR for_IF vectors_NN2 )_) be_VBI independent_JJ variables_NN2 ._. 
Finally_RR ,_, when_CS sets_NN2 of_IO differential_JJ equations_NN2 are_VBR written_VVN in_II matrix_NN1 form_NN1 ,_, some_DD of_IO the_AT matrices_NN2 will_VM be_VBI ,_, in_II effect_NN1 ,_, operators_NN2 ._. 
We_PPIS2 now_RT proceed_VV0 to_TO examine_VVI some_DD simple_JJ cases_NN2 ._. 
Suppose_VV0 first_MD that_CST we_PPIS2 have_VH0 a_AT1 set_NN1 of_IO variables_NN2 &lsqb;_( formula_NN1 &rsqb;_) related_VVD to_II a_AT1 second_MD set_NN1 &lsqb;_( formula_NN1 &rsqb;_) by_II an_AT1 equation_NN1 of_IO the_AT type_NN1 y_ZZ1 =_FO Ax_NP1 ;_; such_DA a_AT1 relation_NN1 might_VM ,_, for_REX21 example_REX22 ,_, connect_VV0 the_AT coordinates_NN2 ,_, referred_VVN to_II two_MC different_JJ sets_NN2 of_IO axes_NN2 ,_, of_IO a_AT1 point_NN1 moving_VVG in_II n-space_NN1 ._. 
Such_DA a_AT1 relation_NN1 is_VBZ called_VVN liner_NN1 substitution_NN1 ._. 
Apart_II21 from_II22 its_APPGE intrinsic_JJ importance_NN1 ,_, it_PPH1 is_VBZ of_IO interest_NN1 since_CS the_AT subject_NN1 of_IO matrices_NN2 evolved_VVN from_II it_PPH1 ._. 
Cayley_NP1 took_VVD a_AT1 set_NN1 of_IO equations_NN2 such_II21 as_II22 &lsqb;_( formula_NN1 &rsqb;_) and_CC enclosed_VVD each_DD1 side_NN1 in_II brackets_NN2 ,_, so_CS21 that_CS22 two_MC equal_JJ columns_NN2 could_VM be_VBI connected_VVN by_II a_AT1 single_JJ equality_NN1 sign_NN1 ._. 
He_PPHS1 then_RT extracted_VVD the_AT vector_NN1 x_ZZ1 and_CC wrote_VVD it_PPH1 vertically_RR as_II a_AT1 parallel_NN1 to_II y_ZZ1 ,_, so_RR requiring_VVG the_AT row-by-column_JJ rule_NN1 for_IF multiplication_NN1 ._. 
A_AT1 second_MD substitution_NN1 of_IO the_AT same_DA kind_NN1 ,_, such_DA as_CSA z_ZZ1 =_FO By._NP1 implies_VVZ z_ZZ1 =_FO By_II =_FO BAx_NP1 ,_, so_CS21 that_CS22 BA_NP1 is_VBZ the_AT matrix_NN1 converting_VVG x_ZZ1 into_II z_ZZ1 ._. 
If_CS the_AT values_NN2 of_IO x_ZZ1 are_VBR unknown_JJ ,_, but_CCB y_ZZ1 is_VBZ given_VVN (_( say_VV0 y_ZZ1 =_FO b_ZZ1 )_) then_RT we_PPIS2 have_VH0 a_AT1 set_NN1 of_IO linear_JJ algebraic_JJ equations_NN2 for_IF x_ZZ1 :_: Ax_NP1 =_FO b_ZZ1 ,_, of_IO which_DDQ the_AT formal_JJ solution_NN1 is_VBZ &lsqb;_( formula_NN1 &rsqb;_) :_: however_RR ,_, various_JJ well_RR known_VVN methods_NN2 exist_VV0 for_IF finding_VVG x_ZZ1 without_IW computing_NN1 &lsqb;_( formula_NN1 &rsqb;_) (_( see_VV0 2.5_MC )_) ._. 
As_II a_AT1 very_RG simple_JJ example_NN1 of_IO a_AT1 linear_JJ substitution_NN1 ,_, consider_VV0 the_AT coordinates_NN2 of_IO a_AT1 point_NN1 P_ZZ1 relative_II21 to_II22 two_MC sets_NN2 of_IO axes_NN2 &lsqb;_( formula_NN1 &rsqb;_) mutually_RR inclined_JJ at_II an_AT1 angle_NN1 &lsqb;_( formula_NN1 &rsqb;_) Evidently_RR &lsqb;_( formula_NN1 &rsqb;_) elimination_NN1 of_IO r_ZZ1 ,_, &lsqb;_( formula_NN1 &rsqb;_) gives_VVZ &lsqb;_( formula_NN1 &rsqb;_) and_CC it_PPH1 should_VM be_VBI noted_VVN that_CST in_II this_DD1 case_NN1 the_AT matrix_NN1 of_IO transformation_NN1 is_VBZ orthogonal_JJ ._. 
The_AT formal_JJ solution_NN1 &lsqb;_( formula_NN1 &rsqb;_) (_( 3_MC )_) assumes_VVZ that_CST A_ZZ1 is_VBZ non-singular_JJ ._. 
Suppose_VV0 ,_, however_RR ,_, that_CST A_ZZ1 has_VHZ degeneracy_NN1 s_ZZ1 ._. 
Then_RT s_ZZ1 of_IO the_AT rows_NN2 implicit_JJ in_II (_( 3_MC )_) may_VM be_VBI compounded_VVN from_II the_AT remaining_JJ n_ZZ1 s_ZZ1 rows_NN2 ,_, and_CC so_RR provide_VV0 no_AT extra_JJ information_NN1 ._. 
These_DD2 s_ZZ1 rows_NN2 ,_, including_II the_AT appropriate_JJ elements_NN2 of_IO b_ZZ1 ,_, may_VM thus_RR be_VBI discarded_VVN ._. 
We_PPIS2 are_VBR left_VVN with_IW n_ZZ1 s_ZZ1 rows_NN2 which_DDQ may_VM be_VBI written_VVN in_II partitioned_JJ matrix_NN1 form_NN1 as_CSA &lsqb;_( formula_NN1 &rsqb;_) where_RRQ A1_FO is_VBZ a_AT1 non-singular_JJ submatrix_NN1 of_IO A_ZZ1 of_IO order_NN1 (_( n_ZZ1 s_ZZ1 )_) (_( some_DD reordering_NN1 of_IO the_AT elements_NN2 of_IO x_ZZ1 may_VM be_VBI necessary_JJ to_TO obtain_VVI the_AT non-singular_JJ minor_JJ )_) ,_, &lsqb;_( formula_NN1 &rsqb;_) is_VBZ of_IO the_AT order_NN1 &lsqb;_( formula_NN1 &rsqb;_) and_CC c_ZZ1 is_VBZ the_AT reduced_JJ vector_NN1 b_ZZ1 ;_; x_ZZ1 has_VHZ been_VBN partitioned_VVN conformably_RR into_II n_ZZ1 s_ZZ1 elements_NN2 (_( y_ZZ1 )_) and_CC s_ZZ1 elements_NN2 (_( z_ZZ1 )_) ._. 
We_PPIS2 may_VM write_VVI this_DD1 equation_NN1 as_CSA &lsqb;_( formula_NN1 &rsqb;_) Equation_NN1 (_( 5_MC )_) is_VBZ known_VVN as_II a_AT1 parametric_JJ solution_NN1 :_: since_CS we_PPIS2 have_VH0 only_RR n_ZZ1 s_ZZ1 independent_JJ relations_NN2 we_PPIS2 can_VM determine_VVI only_JJ n_ZZ1 s_ZZ1 unknowns_NN2 (_( y_ZZ1 )_) in_II31 terms_II32 of_II33 the_AT remaining_JJ s_ZZ1 quantities_NN2 (_( z_ZZ1 )_) ,_, which_DDQ may_VM be_VBI regarded_VVN as_CSA arbitrary_JJ parameters_NN2 in_II the_AT problem._NNU 1.11_MC BILINEAR_JJ AND_CC QUADRATIC_NP1 FORMS_VVZ Scalar_JJ functions_NN2 often_RR arise_VV0 which_DDQ are_VBR linear_JJ and_CC homogeneous_JJ in_II two_MC sets_NN2 of_IO variables_NN2 x1_FO ,_, ..._... ,_, xn_FO and_CC y1_FO ,_, ..._... ,_, ym_NN1 ._. 
By_II inspection_NN1 ,_, it_PPH1 is_VBZ evident_JJ that_CST such_DA a_AT1 function_NN1 may_VM be_VBI written_VVN as_CSA &lsqb;_( formula_NN1 &rsqb;_) when_RRQ x_ZZ1 ,_, y_ZZ1 are_VBR written_VVN as_CSA columns_NN2 ._. 
Here_RL A_ZZ1 ,_, of_IO order_NN1 (_( m_ZZ1 n_ZZ1 )_) is_VBZ called_VVN the_AT matrix_NN1 of_IO the_AT form_NN1 ._. 
Suppose_VV0 we_PPIS2 wish_VV0 to_TO express_VVI f_ZZ1 in_II31 terms_II32 of_II33 new_JJ variables_NN2 &lsqb;_( formula_NN1 &rsqb;_) ,_, where_RRQ &lsqb;_( formula_NN1 &rsqb;_) then_RT &lsqb;_( formula_NN1 &rsqb;_) so_CS21 that_CS22 &lsqb;_( formula_NN1 &rsqb;_) is_VBZ the_AT matrix_NN1 of_IO the_AT new_JJ form_NN1 ._. 
We_PPIS2 say_VV0 hat_NN1 A_ZZ1 and_CC &lsqb;_( formula_NN1 &rsqb;_) are_VBR connected_VVN by_II an_AT1 equivalent_JJ transformation_NN1 ._. 
If_CS the_AT elements_NN2 of_IO x_ZZ1 ,_, y_ZZ1 are_VBR independent_JJ variables_NN2 ,_, we_PPIS2 may_VM differentiate_VVI with_II31 respect_II32 to_II33 them_PPHO2 ._. 
For_REX21 example_REX22 ,_, from_II (_( 1_MC1 )_) above_RL ,_, &lsqb;_( formula_NN1 &rsqb;_) is_VBZ the_AT first_MD element_NN1 in_II the_AT column_NN1 Ax_NP1 ,_, &lsqb;_( formula_NN1 &rsqb;_) the_AT second_NNT1 ,_, etc._RA ,_, so_CS21 that_CS22 &lsqb;_( formula_NN1 &rsqb;_) In_II this_DD1 text_NN1 ,_, we_PPIS2 shall_VM be_VBI more_RGR concerned_JJ with_IW quadratic_JJ forms_NN2 ,_, obtained_VVN from_II bilinear_JJ forms_NN2 such_II21 as_II22 &lsqb;_( formula_NN1 &rsqb;_) when_RRQ y_ZZ1 =_FO x_ZZ1 ,_, so_CS21 that_CS22 &lsqb;_( formula_NN1 &rsqb;_) where_RRQ A_ZZ1 is_VBZ now_RT necessarily_RR square_JJ ._. 
Moreover_RR ,_, although_CS in_II the_AT formulation_NN1 of_IO A_ZZ1 the_AT elements_NN2 Aij_VV0 and_CC Aji_NN1 may_VM not_XX be_VBI equal_JJ ,_, we_PPIS2 may_VM note_VVI that_CST (_( see_VV0 1.2_MC )_) &lsqb;_( formula_NN1 &rsqb;_) and_CC on_II transposition_NN1 it_PPH1 follows_VVZ that_CST &lsqb;_( formula_NN1 &rsqb;_) so_CS21 that_CS22 A_ZZ1 may_VM be_VBI replaced_VVN by_II its_APPGE symmetric_JJ equivalent_NN1 &lsqb;_( formula_NN1 &rsqb;_) ;_; we_PPIS2 now_RT assume_VV0 that_CST A_ZZ1 is_VBZ symmetric_JJ ._. 
Then_RT if_CS we_PPIS2 form_VV0 &lsqb;_( formula_NN1 &rsqb;_) we_PPIS2 have_VH0 &lsqb;_( formula_NN1 &rsqb;_) since_CS transposition_NN1 of_IO the_AT second_MD term_NN1 repeats_VVZ the_AT first_MD ._. 
Thus_RR ,_, since_CS &lsqb;_( formula_NN1 &rsqb;_) If_CS we_PPIS2 express_VV0 a_AT1 quadratic_JJ form_NN1 &lsqb;_( formula_NN1 &rsqb;_) in_II31 terms_II32 of_II33 another_DD1 set_NN1 of_IO variables_NN2 y_ZZ1 given_VVN by_II the_AT linear_JJ substitution_NN1 x_ZZ1 =_FO By_II ,_, then_RT &lsqb;_( formula_NN1 &rsqb;_) where_RRQ &lsqb;_( formula_NN1 &rsqb;_) and_CC we_PPIS2 observe_VV0 that_CST C_ZZ1 ,_, like_II A_ZZ1 ,_, is_VBZ symmetric_JJ ._. 
A_AT1 transformation_NN1 of_IO type_NN1 (_( 2_MC )_) ,_, in_II which_DDQ A_ZZ1 ,_, C_ZZ1 need_VM not_XX in_RR21 general_RR22 be_VBI symmetric_JJ ,_, but_CCB in_II which_DDQ B_ZZ1 is_VBZ square_JJ and_CC non-singular_NN1 ,_, is_VBZ described_VVN as_II a_AT1 congruent_JJ transformation_NN1 and_CC is_VBZ of_IO particular_JJ importance_NN1 in_II dynamics_NN ._. 
A_AT1 quadratic_JJ form_NN1 &lsqb;_( formula_NN1 &rsqb;_) and_CC the_AT matrix_NN1 A_ZZ1 within_II f_ZZ1 ,_, are_VBR said_VVN to_TO be_VBI positive_JJ definite_JJ (_( pos._NNU def._NNU )_) if_CS &lsqb;_( formula_NN1 &rsqb;_) for_IF all_DB real_JJ ,_, non-null_JJ vectors_NN2 x_ZZ1 ._. 
We_PPIS2 may_VM construct_VVI a_AT1 table_NN1 for_IF such_DA definitions_NN2 (_( Table_NN1 1_MC1 )_) ._. 
Necessary_JJ and_CC sufficient_JJ conditions_NN2 for_IF a_AT1 quadratic_JJ form_NN1 to_TO be_VBI pos._NNU def._NNU are_VBR established_VVN in_II Theorem_NN1 XIII_MC of_IO 1.22_MC ._. 
If_CS a_AT1 skew-symmetric_JJ ,_, transposition_NN1 shows_VVZ that_CST &lsqb;_( formula_NN1 &rsqb;_) vanishes._NNU &lsqb;_( formula_NN1 &rsqb;_) 1.12_MC EQUIVALENT_JJ MATRICES_NN2 AND_CC CANONICAL_JJ FORMS_NN2 We_PPIS2 have_VH0 just_RR defined_VVN the_AT general_JJ equivalent_JJ transformation_NN1 ,_, and_CC this_DD1 leads_VVZ naturally_RR to_II the_AT important_JJ concept_NN1 of_IO equivalent_JJ matrices_NN2 ._. 
We_PPIS2 approach_VV0 this_DD1 by_II considering_VVG the_AT elementary_JJ operations_NN2 which_DDQ are_VBR commonly_RR used_VVN in_II the_AT condensation_NN1 of_IO determinants_NN2 ._. 
Let_VV0 I_ZZ1 &lsqb;_( formula_NN1 &rsqb;_) be_VBI the_AT unit_NN1 matrix_NN1 with_IW its_APPGE ith_MD and_CC jth_MD rows_NN2 interchanged_VVD ._. 
The_AT new_JJ matrix_NN1 is_VBZ symmetric_JJ ,_, since_CS both_DB2 its_APPGE i_ZZ1 ,_, jth_MD and_CC j_ZZ1 ,_, ith_MD elements_NN2 are_VBR unity_NN1 ;_; moreover_RR ,_, its_APPGE determinant_NN1 is_VBZ -1_MC ,_, since_CS in_II moving_VVG the_AT jth_MD row_NN1 to_II the_AT ith_MD position_NN1 &lsqb;_( formula_NN1 &rsqb;_) we_PPIS2 cross_VV0 j_ZZ1 i_ZZ1 rows_NN2 ;_; but_CCB the_AT original_JJ ith_MD row_NN1 is_VBZ now_RT the_AT i_ZZ1 +_FO 1th_FO and_CC so_RR in_II taking_VVG it_PPH1 to_II the_AT jth_MD position_NN1 we_PPIS2 cross_VV0 j_ZZ1 i_ZZ1 1_MC1 rows_NN2 ._. 
In_II31 view_II32 of_II33 its_APPGE symmetry_NN1 &lsqb;_( formula_NN1 &rsqb;_) can_VM equally_RR be_VBI obtained_VVN by_II interchange_NN1 of_IO the_AT ith_MD and_CC jth_MD columns_NN2 instead_II21 of_II22 rows_NN2 ._. 
Let_VV0 us_PPIO2 restrict_VVI ourselves_PPX2 to_TO square_VVI matrices_NN2 ;_; then_RT premultiplication_NN1 of_IO any_DD matrix_NN1 A_ZZ1 by_II &lsqb;_( formula_NN1 &rsqb;_) has_VHZ the_AT effect_NN1 of_IO interchanging_VVG the_AT ith_MD and_CC jth_MD rows_NN2 of_IO A_ZZ1 ;_; postmultiplication_NN1 of_IO A_ZZ1 by_II &lsqb;_( formula_NN1 &rsqb;_) interchanges_NN2 the_AT ith_MD and_CC jth_MD columns_NN2 ._. 
Next_MD ,_, let_VV0 &lsqb;_( formula_NN1 &rsqb;_) be_VBI the_AT unit_NN1 matrix_NN1 with_IW the_AT addition_NN1 of_IO an_AT1 element_NN1 k_ZZ1 in_II the_AT &lsqb;_( formula_NN1 &rsqb;_) position_NN1 ._. 
Then_RT premultiplication_NN1 of_IO A_ZZ1 by_II &lsqb;_( formula_NN1 &rsqb;_) has_VHZ the_AT effect_NN1 of_IO adding_VVG k_ZZ1 times_NNT2 the_AT jth_MD row_NN1 of_IO A_ZZ1 to_II the_AT ith_MD row_NN1 ;_; similarly_RR the_AT operation_NN1 AI(kij)_NP1 has_VHZ the_AT effect_NN1 of_IO adding_VVG k_ZZ1 times_NNT2 the_AT ith_MD column_NN1 of_IO A_ZZ1 to_II the_AT jth_MD column_NN1 ._. 
The_AT determinant_NN1 of_IO &lsqb;_( formula_NN1 &rsqb;_) is_VBZ clearly_RR unity_NN1 ._. 
Finally_RR ,_, let_VV0 &lsqb;_( formula_NN1 &rsqb;_) be_VBI the_AT unit_NN1 matrix_NN1 with_IW the_AT element_NN1 &lsqb;_( formula_NN1 &rsqb;_) substituted_VVD for_IF the_AT unit_NN1 in_II the_AT i_ZZ1 ,_, ith_MD position_NN1 ._. 
Then_RT I(li)A_NP1 multiplies_VVZ the_AT ith_MD row_NN1 of_IO A_ZZ1 by_II l_ZZ1 ;_; A&lsqb;formula&rsqb;_NP1 multiplies_VVZ the_AT ith_MD column_NN1 of_IO A_ZZ1 by_II l_ZZ1 ._. 
This_DD1 operation_NN1 is_VBZ evidently_RR an_AT1 extension_NN1 of_IO the_AT I&lsqb;formula&rsqb;_JJ operation_NN1 ;_; instead_II21 of_II22 adding_VVG multiples_NN2 of_IO a_AT1 different_JJ row_NN1 (_( or_CC column_NN1 )_) to_II a_AT1 given_JJ row_NN1 ,_, it_PPH1 adds_VVZ multiples_NN2 of_IO the_AT same_DA row_NN1 ;_; however_RR ,_, it_PPH1 differs_VVZ in_II that_DD1 now_RT the_AT determinant_NN1 is_VBZ l_ZZ1 ._. 
The_AT reciprocal_JJ of_IO I&lsqb;formula&rsqb;_NP1 is_VBZ I&lsqb;formula&rsqb;_NP1 itself_PPX1 ,_, since_CS a_AT1 double_JJ interchange_NN1 of_IO two_MC rows_NN2 restores_VVZ the_AT original_JJ matrix_NN1 ;_; the_AT reciprocal_JJ of_IO I&lsqb;formula&rsqb;_NP1 is_VBZ also_RR evidently_RR I&lsqb;formula&rsqb;_NP1 and_CC that_DD1 of_IO &lsqb;_( formula_NN1 &rsqb;_) ._. 
Any_DD product_NN1 of_IO these_DD2 various_JJ matrices_NN2 is_VBZ always_RR non-singular_JJ ._. 
Two_MC matrices_NN2 ,_, A_ZZ1 ,_, B_ZZ1 are_VBR said_VVN to_TO be_VBI equivalent_JJ if_CS one_PN1 can_VM be_VBI derived_VVN from_II the_AT other_JJ by_II any_DD finite_JJ number_NN1 of_IO elementary_JJ operations_NN2 of_IO the_AT types_NN2 specified_VVN above_RL ._. 
Hence_RR ,_, if_CS all_DB the_AT premultiplications_NN2 and_CC postmultiplications_NN2 are_VBR grouped_VVN together_RL to_TO form_VVI (_( non-singular_NN1 )_) matrices_NN2 P_ZZ1 and_CC Q_ZZ1 respectively_RR ,_, and_CC A_ZZ1 ,_, B_ZZ1 are_VBR equivalent_JJ ,_, then_RT B_ZZ1 =_FO PAQ_NP1 and_CC &lsqb;_( formula_NN1 &rsqb;_) Since_CS k_ZZ1 ,_, l_ZZ1 are_VBR arbitrary_JJ (_( except_CS21 that_CS22 &lsqb;_( formula_NN1 &rsqb;_) )_) it_PPH1 follows_VVZ that_CST there_EX is_VBZ a_AT1 infinite_JJ number_NN1 of_IO matrices_NN2 B_ZZ1 equivalent_JJ to_II A._NP1 However_RR ,_, there_EX is_VBZ one_MC1 matrix_NN1 which_DDQ is_VBZ of_IO especially_RR simple_JJ form_NN1 ;_; it_PPH1 is_VBZ described_VVN as_II the_AT canonical_JJ form_NN1 of_IO A._NNU First_MD ,_, let_VV0 us_PPIO2 suppose_VVI that_CST A_ZZ1 is_VBZ non-singular_JJ ._. 
If_CS A11_FO =_FO 0_MC we_PPIS2 can_VM bring_VVI any_DD non-zero_JJ element_NN1 to_II the_AT A11_FO position_NN1 by_II the_AT use_NN1 of_IO I&lsqb;formula&rsqb;_NP1 we_PPIS2 can_VM then_RT reduce_VVI this_DD1 element_NN1 to_II unity_NN1 by_II the_AT use_NN1 of_IO I(l)_FW ,_, and_CC then_RT reduce_VV0 all_DB the_AT remaining_JJ elements_NN2 of_IO the_AT first_MD row_NN1 and_CC the_AT first_MD column_NN1 to_TO zero_VVI using_VVG I(k)_NN1 ._. 
We_PPIS2 do_VD0 the_AT same_DA for_IF the_AT submatrix_NN1 bordered_VVN by_II the_AT first_MD row_NN1 and_CC column_NN1 ,_, and_RR31 so_RR32 on_RR33 ,_, so_CS21 that_CS22 finally_RR we_PPIS2 reach_VV0 the_AT unit_NN1 matrix_NN1 as_II the_AT canonical_JJ form_NN1 for_IF A._NP1 It_PPH1 is_VBZ to_TO be_VBI observed_VVN that_CST ,_, since_CS in_II using_VVG elementary_JJ operations_NN2 I_ZZ1 (_( i_ZZ1 ,_, j_ZZ1 )_) ,_, I(k)_NP1 ,_, I(l)_FW on_II A_ZZ1 we_PPIS2 are_VBR effectively_RR at_II each_DD1 stage_NN1 multiplying_VVG together_RL determinants_NN2 neither_DD1 of_IO which_DDQ vanishes_VVZ ,_, the_AT reduction_NN1 of_IO A_ZZ1 to_II I_PPIS1 must_VM be_VBI possible_JJ when_CS A_ZZ1 is_VBZ on-singular_JJ ._. 
In_II passing_NN1 ,_, we_PPIS2 may_VM note_VVI that_DD1 ,_, since_CS PAQ_NP1 =_FO I_ZZ1 ,_, &lsqb;_( formula_NN1 &rsqb;_) Now_RT suppose_VV0 A_ZZ1 to_TO be_VBI singular_JJ ,_, of_IO rank_NN1 r_ZZ1 and_CC order_NN1 n_ZZ1 ._. 
This_DD1 means_VVZ that_CST all_DB minor_JJ determinants_NN2 of_IO order_NN1 r_ZZ1 +_FO 1_MC1 vanish_VV0 ,_, but_CCB at_RR21 least_RR22 one_MC1 minor_NN1 of_IO order_NN1 r_ZZ1 is_VBZ not_XX zero_NN1 ._. 
None_PN of_IO the_AT operations_NN2 I_ZZ1 (_( i_ZZ1 ,_, j_ZZ1 )_) ,_, I(k)_NP1 ,_, I(l)_FW ,_, in_CS41 so_CS42 far_CS43 as_CS44 they_PPHS2 affect_VV0 the_AT minors_NN2 ,_, can_VM make_VVI a_AT1 minor_NN1 of_IO order_NN1 r_ZZ1 +_FO 1_MC1 nonzero_NN1 ,_, nor_CC make_VV0 the_AT minor_NN1 of_IO order_NN1 r_ZZ1 vanish_VV0 ,_, since_CS they_PPHS2 merely_RR condense_VV0 these_DD2 minors_NN2 ._. 
We_PPIS2 may_VM therefore_RR use_VVI the_AT processes_NN2 above_RL as_II21 for_II22 a_AT1 non-singular_JJ matrix_NN1 ;_; we_PPIS2 achieve_VV0 a_AT1 series_NN of_IO units_NN2 in_II the_AT &lsqb;_( formula_NN1 &rsqb;_) positions_NN2 ;_; the_AT other_JJ elements_NN2 in_II the_AT matrix_NN1 are_VBR all_DB zero_MC ._. 
The_AT canonical_JJ form_NN1 of_IO A_ZZ1 is_VBZ thus_RR &lsqb;_( formula_NN1 &rsqb;_) where_RRQ I_ZZ1 is_VBZ of_IO order_NN1 r_ZZ1 ._. 
If_CS B_ZZ1 =_FO PAQ_NP1 ,_, where_CS P_ZZ1 ,_, Q_ZZ1 are_VBR non-singular_JJ but_CCB otherwise_RR arbitrary_JJ ,_, then_RT the_AT canonical_JJ forms_NN2 for_IF P_ZZ1 ,_, Q_ZZ1 are_VBR the_AT unit_NN1 matrix_NN1 ._. 
It_PPH1 follows_VVZ that_CST P_ZZ1 ,_, Q_ZZ1 can_VM be_VBI constructed_VVN from_II I_MC1 by_II elementary_JJ operations_NN2 ;_; accordingly_RR ,_, if_CS any_DD such_DA product_NN1 ,_, provided_CS only_RR that_CST P_ZZ1 ,_, Q_ZZ1 are_VBR non-singular_JJ ,_, A_ZZ1 and_CC B_ZZ1 are_VBR equivalent_JJ matrices_NN2 ;_; and_CC in_RR21 particular_RR22 ,_, they_PPHS2 will_VM have_VHI the_AT same_DA rank._NNU 1.13_MC LAMBDA-MATRICES_NN2 We_PPIS2 have_VH0 already_RR defined_VVN a_AT1 lambda-matrix_NN1 of_IO which_DDQ the_AT elements_NN2 are_VBR rational_JJ integral_JJ functions_NN2 of_IO some_DD parameter_NN1 &lsqb;_( formula_NN1 &rsqb;_) Such_DA matrices_NN2 arise_VV0 in_II many_DA2 problems_NN2 of_IO mechanics_NN2 ._. 
For_REX21 example_REX22 ,_, suppose_VV0 we_PPIS2 have_VH0 a_AT1 pair_NN of_IO simultaneous_JJ differential_JJ equations_NN2 &lsqb;_( formula_NN1 &rsqb;_) where_RRQ the_AT coefficients_NN2 &lsqb;_( formula_NN1 &rsqb;_) are_VBR constants_NN2 ._. 
We_PPIS2 can_VM write_VVI these_DD2 as_CSA &lsqb;_( formula_NN1 &rsqb;_) where_RRQ the_AT column_NN1 matrices_NN2 are_VBR all_DB functions_NN2 of_IO time_NNT1 ._. 
An_AT1 alternative_JJ symbolic_JJ formulation_NN1 would_VM be_VBI &lsqb;_( formula_NN1 &rsqb;_) where_RRQ D_ZZ1 =_FO d/dt_FU and_CC the_AT square_JJ matrix_NN1 is_VBZ now_RT an_AT1 operator_NN1 on_II x_ZZ1 ._. 
Finally_RR ,_, if_CS we_PPIS2 are_VBR seeking_VVG the_AT complementary_JJ function_NN1 of_IO the_AT equations_NN2 ,_, we_PPIS2 write_VV0 x_ZZ1 proportional_JJ to_II exp_JJ &lsqb;_( formula_NN1 &rsqb;_) and_CC obtain_VV0 &lsqb;_( formula_NN1 &rsqb;_) as_II the_AT equations_NN2 determining_VVG &lsqb;_( formula_NN1 &rsqb;_) and_CC the_AT ratio_NN1 &lsqb;_( formula_NN1 &rsqb;_) In_II abbreviated_JJ form_NN1 this_DD1 can_VM be_VBI written_VVN &lsqb;_( formula_NN1 &rsqb;_) ._. 
The_AT equation_NN1 determining_VVG &lsqb;_( formula_NN1 &rsqb;_) is_VBZ obtained_VVN from_II the_AT vanishing_JJ of_IO the_AT determinant_NN1 of_IO the_AT lambda-matrix_NN1 ;_; it_PPH1 is_VBZ evidently_RR a_AT1 quartic_JJ in_II &lsqb;_( formula_NN1 &rsqb;_) and_CC each_DD1 of_IO the_AT four_MC roots_NN2 will_VM have_VHI its_APPGE own_DA associated_JJ vector_NN1 x_ZZ1 ._. 
While_CS it_PPH1 is_VBZ often_RR convenient_JJ to_TO deal_VVI with_IW equations_NN2 such_II21 as_II22 (_( 2_MC )_) in_II the_AT way_NN1 indicated_VVD ,_, it_PPH1 is_VBZ also_RR often_RR useful_JJ to_TO transform_VVI them_PPHO2 into_II first-order_JJ equations_NN2 ._. 
This_DD1 may_VM be_VBI done_VDN in_II various_JJ ways_NN2 ,_, but_CCB one_PN1 simple_JJ and_CC useful_JJ device_NN1 is_VBZ to_TO adopt_VVI two_MC auxiliary_JJ variables_NN2 :_: &lsqb;_( formula_NN1 &rsqb;_) and_CC to_TO write_VVI the_AT equations_NN2 in_II the_AT form_NN1 &lsqb;_( formula_NN1 &rsqb;_) Then_RT if_CS we_PPIS2 now_RT write_VV0 &lsqb;_( formula_NN1 &rsqb;_) and_CC &lsqb;_( formula_NN1 &rsqb;_) proportional_JJ to_II exp_JJ &lsqb;_( formula_NN1 &rsqb;_) and_CC again_RT seek_VV0 the_AT complementary_JJ function_NN1 ,_, the_AT four_MC equations_NN2 may_VM be_VBI written_VVN ,_, with_IW a_AT1 little_JJ rearrangement_NN1 ,_, &lsqb;_( formula_NN1 &rsqb;_) This_DD1 equation_NN1 can_VM clearly_RR be_VBI written_VVN in_II partitioned_JJ form_NN1 as_CSA &lsqb;_( formula_NN1 &rsqb;_) or_CC &lsqb;_( formula_NN1 &rsqb;_) where_RRQ &lsqb;_( formula_NN1 &rsqb;_) The_AT quartic_JJ determinantal_JJ equation_NN1 obtained_VVN from_II this_DD1 first-order_JJ set_NN1 of_IO equations_NN2 may_VM be_VBI seen_VVN by_II inspection_NN1 to_TO be_VBI identical_JJ with_IW that_DD1 obtained_VVN from_II the_AT quadratic_JJ formulation_NN1 ._. 
A_AT1 matrix_NN1 of_IO the_AT form_NN1 M_ZZ1 +_FO &lsqb;_( formula_NN1 &rsqb;_) is_VBZ called_VVN a_AT1 matrix_NN1 pencil_NN1 (_( see_VV0 2.9_MC ,_, 2.10_MC )_) ._. 
1.14_MC ADJUGATE_NN1 LAMBDA-MATRICES_NN2 If_CS A_ZZ1 &lsqb;_( formula_NN1 &rsqb;_) is_VBZ a_AT1 lambda-matrix_NN1 ,_, its_APPGE elements_NN2 are_VBR rational_JJ integral_JJ functions_NN2 of_IO gl_NNU and_CC so_RG also_RR ,_, therefore_RR ,_, are_VBR the_AT cofactors_NN2 ga_NN1 of_IO the_AT elements_NN2 of_IO A_ZZ1 :_: hence_RR the_AT adjugate_NN1 ,_, &lsqb;_( formula_NN1 &rsqb;_) of_IO A_ZZ1 is_VBZ also_RR a_AT1 lambda-matrix_NN1 ,_, and_CC &lsqb;_( formula_NN1 &rsqb;_) We_PPIS2 shall_VM denote_VVI the_AT determinant_NN1 &lsqb;_( formula_NN1 &rsqb;_) by_II &lsqb;_( formula_NN1 &rsqb;_) ._. 
The_AT reciprocal_JJ of_IO A_ZZ1 ,_, however_RR ,_, being_NN1 &lsqb;_( formula_NN1 &rsqb;_) divided_VVN by_II &lsqb;_( formula_NN1 &rsqb;_) is_VBZ not_XX a_AT1 lambda-matrix_NN1 ,_, unless_CS it_PPH1 should_VM happen_VVI that_CST &lsqb;_( formula_NN1 &rsqb;_) is_VBZ independent_JJ of_IO &lsqb;_( formula_NN1 &rsqb;_) ._. 
1.15_MC EQUIVALENT_JJ LAMBDA-MATRICES_NN2 If_CS we_PPIS2 have_VH0 an_AT1 equivalent_JJ transformation_NN1 &lsqb;_( formula_NN1 &rsqb;_) where_RRQ P_ZZ1 ,_, Q_ZZ1 may_VM be_VBI constructed_VVN by_II any_DD of_IO the_AT elementary_JJ operations_NN2 defined_VVN in_II 2.12_MC ,_, then_RT B_ZZ1 and_CC A_ZZ1 are_VBR equivalent_JJ ._. 
It_PPH1 should_VM be_VBI noted_VVN that_CST the_AT determinants_NN2 of_IO P_ZZ1 ,_, Q_ZZ1 must_VM be_VBI non-vanishing_JJ constants_NN2 ,_, though_CS &lsqb;_( formula_NN1 &rsqb;_) may_VM appear_VVI in_II them_PPHO2 ._. 
This_DD1 implies_VVZ that_CST in_II an_AT1 operation_NN1 I(k)_NN1 we_PPIS2 may_VM see_VVI I_MC1 (_( &lsqb;_( formula_NN1 &rsqb;_) )_) ,_, ;_; but_CCB &lsqb;_( formula_NN1 &rsqb;_) will_VM not_XX appear_VVI in_II I(l)_FW operations_NN2 ,_, except_CS in_II a_AT1 self-cancelling_JJ form_NN1 ,_, e.g._REX we_PPIS2 could_VM use_VVI I(l)_FW with_IW &lsqb;_( formula_NN1 &rsqb;_) in_II the_AT ith_MD place_NN1 ,_, following_VVG with_IW I(l)_FW with_IW &lsqb;_( formula_NN1 &rsqb;_) in_II the_AT jith_NN1 place_NN1 ,_, so_CS21 that_CS22 the_AT product_NN1 of_IO the_AT two_MC steps_NN2 has_VHZ unit_NN1 determinant_NN1 ._. 
I_ZZ1 (_( i_ZZ1 ,_, j_ZZ1 )_) is_VBZ ,_, of_RR21 course_RR22 ,_, independent_JJ of_IO &lsqb;_( formula_NN1 &rsqb;_) ._. 
Using_VVG elementary_JJ operations_NN2 of_IO this_DD1 kind_NN1 ,_, we_PPIS2 may_VM construct_VVI a_AT1 canonical_JJ form_NN1 of_IO lambda-matrices_NN2 ,_, which_DDQ is_VBZ diagonal_JJ ,_, the_AT elements_NN2 in_II the_AT diagonal_JJ being_NN1 functions_NN2 of_IO &lsqb;_( formula_NN1 &rsqb;_) ._. 
The_AT canonical_JJ form_NN1 is_VBZ not_XX unique_JJ ;_; the_AT best-known_JJ form_NN1 is_VBZ that_CST due_II21 to_II22 Smith_NP1 (_( see_VV0 Ref._NN1 (_( 1_MC1 )_) )_) ._. 
If_CS C_ZZ1 (_( &lsqb;_( formula_NN1 &rsqb;_) )_) ;_; is_VBZ the_AT canonical_JJ form_NN1 ,_, since_CS C_ZZ1 (_( &lsqb;_( formula_NN1 &rsqb;_) )_) =_FO PA_NNU (_( &lsqb;_( formula_NN1 &rsqb;_) )_) Q_ZZ1 ;_; we_PPIS2 have_VH0 &lsqb;_( formula_NN1 &rsqb;_) and_CC since_CS &lsqb;_( formula_NN1 &rsqb;_) are_VBR non-vanishing_JJ constants_NN2 ,_, we_PPIS2 have_VH0 that_DD1 &lsqb;_( formula_NN1 &rsqb;_) and_CC &lsqb;_( formula_NN1 &rsqb;_) differ_VV0 only_RR by_II a_AT1 scalar_JJ multiplier_NN1 ._. 
Hence_RR the_AT roots_NN2 of_IO &lsqb;_( formula_NN1 &rsqb;_) are_VBR identical_JJ with_IW those_DD2 of_IO &lsqb;_( formula_NN1 &rsqb;_) ._. 
1.16_MC THE_AT CHARACTERISTIC_JJ MATRIX_NN1 AND_CC ITS_APPGE EIGENVALUES_NN2 Suppose_VV0 we_PPIS2 have_VH0 a_AT1 linear_JJ substitution_NN1 Ax_NP1 =_FO y_ZZ1 in_II which_DDQ it_PPH1 is_VBZ required_VVN that_CST y_ZZ1 be_VBI a_AT1 scalar_JJ multiple_NN1 of_IO x_ZZ1 ,_, say_VV0 &lsqb;_( formula_NN1 &rsqb;_) x_ZZ1 ._. 
Then_RT &lsqb;_( formula_NN1 &rsqb;_) x_ZZ1 Ax_NP1 =_FO (_( &lsqb;_( formula_NN1 &rsqb;_) I_ZZ1 A_ZZ1 )_) x_ZZ1 =_FO 0_MC ._. 
Then_RT the_AT square_JJ lambda-matrix_NN1 (_( &lsqb;_( formula_NN1 &rsqb;_) I_ZZ1 A_ZZ1 )_) ,_, which_DDQ is_VBZ the_AT characteristic_JJ matrix_NN1 of_IO A_ZZ1 (_( see_VV0 1.8_MC )_) ,_, must_VM be_VBI singular_JJ ;_; i.e._REX the_AT determinant_NN1 &lsqb;_( formula_NN1 &rsqb;_) if_CS x_ZZ1 is_VBZ not_XX to_TO vanish_VVI ._. 
This_DD1 important_JJ equation_NN1 is_VBZ called_VVN the_AT characteristic_JJ equation_NN1 ._. 
If_CS A_ZZ1 (_( which_DDQ is_VBZ necessarily_RR square_JJ )_) is_VBZ of_IO order_NN1 n_ZZ1 ,_, then_RT clearly_RR the_AT characteristic_JJ equation_NN1 may_VM be_VBI written_VVN &lsqb;_( formula_NN1 &rsqb;_) The_AT n_ZZ1 roots_NN2 of_IO this_DD1 equation_NN1 ,_, say_VV0 &lsqb;_( formula_NN1 &rsqb;_) are_VBR called_VVN eigenvalues_NN2 ._. 
In_II what_DDQ follows_VVZ ,_, we_PPIS2 shall_VM now_RT regard_VVI &lsqb;_( formula_NN1 &rsqb;_) 1_MC1 ,_, ..._... ,_, &lsqb;_( formula_NN1 &rsqb;_) n_ZZ1 as_CSA all_DB different_JJ ._. 
An_AT1 alternative_JJ formulation_NN1 of_IO the_AT characteristic_JJ function_NN1 is_VBZ evidently_RR &lsqb;_( formula_NN1 &rsqb;_) and_CC it_PPH1 is_VBZ to_TO be_VBI observed_VVN that_CST the_AT two_MC formulations_NN2 imply_VV0 ,_, inter_RR21 alia_RR22 ,_, &lsqb;_( formula_NN1 &rsqb;_) and_CC &lsqb;_( formula_NN1 &rsqb;_) where_RRQ Theorems_NN2 II_MC and_CC III_MC of_IO 1.22_MC have_VH0 been_VBN used_VVN ._. 
Let_VV0 us_PPIO2 write_VVI &lsqb;_( formula_NN1 &rsqb;_) then_RT from_II (_( 4_MC )_) ,_, &lsqb;_( formula_NN1 &rsqb;_) so_CS21 that_CS22 if_CS we_PPIS2 regard_VV0 &lsqb;_( formula_NN1 &rsqb;_) 1_MC1 as_CSA typical_JJ of_IO all_DB the_AT roots_NN2 ,_, &lsqb;_( formula_NN1 &rsqb;_) since_CS all_DB the_AT roots_NN2 differ_VV0 ._. 
Now_RT by_II the_AT usual_JJ rule_NN1 for_IF differentiation_NN1 of_IO determinants_NN2 ,_, &lsqb;_( formula_NN1 &rsqb;_) is_VBZ linear_JJ and_CC homogeneous_JJ in_II the_AT first_MD minors_NN2 of_IO gd_JJ (_( &lsqb;_( formula_NN1 &rsqb;_) )_) ._. 
Since_CS ,_, typically_RR ,_, &lsqb;_( formula_NN1 &rsqb;_) it_PPH1 follows_VVZ that_CST not_XX all_DB the_AT first_MD minors_NN2 of_IO &lsqb;_( formula_NN1 &rsqb;_) vanish_VV0 ,_, so_CS21 that_CS22 it_PPH1 is_VBZ simply_RR degenerate_JJ ._. 
For_IF the_AT same_DA reason_NN1 &lsqb;_( formula_NN1 &rsqb;_) ,_, the_AT adjugate_NN1 ,_, is_VBZ not_XX null_JJ ,_, and_CC so_RR has_VHZ unit_NN1 rank_NN1 ,_, as_CSA shown_VVN below_RL ._. 
Since_CS &lsqb;_( formula_NN1 &rsqb;_) vanishes_VVZ but_CCB at_RR21 least_RR22 one_MC1 first_MD minor_NN1 does_VDZ not_XX ,_, there_EX must_VM be_VBI n_ZZ1 1_MC1 independent_JJ columns_NN2 ,_, the_AT remaining_JJ column_NN1 being_VBG expressible_JJ uniquely_RR in_II31 terms_II32 of_II33 the_AT independent_JJ columns_NN2 ._. 
This_DD1 means_VVZ that_CST there_EX will_VM be_VBI a_AT1 vector_NN1 x1_FO ,_, called_VVN an_AT1 eigenvector_NN1 ,_, such_CS21 that_CS22 &lsqb;_( formula_NN1 &rsqb;_) and_CC x1_FO is_VBZ uniquely_RR determined_VVN ,_, apart_II21 from_II22 an_AT1 arbitrary_JJ scalar_JJ multiplier_NN1 ;_; that_REX21 is_REX22 ,_, the_AT ratios_NN2 of_IO the_AT elements_NN2 of_IO x1_FO to_II each_PPX221 other_PPX222 are_VBR uniquely_RR determined_VVN ._. 
Similarly_RR ,_, apart_II21 from_II22 a_AT1 scalar_JJ multiplier_NN1 ,_, there_EX will_VM be_VBI a_AT1 unique_JJ row_NN1 vector_NN1 &lsqb;_( formula_NN1 &rsqb;_) such_CS21 that_CS22 &lsqb;_( formula_NN1 &rsqb;_) It_PPH1 is_VBZ readily_RR shown_VVN that_CST the_AT adjugate_NN1 &lsqb;_( formula_NN1 &rsqb;_) is_VBZ proportional_JJ to_II the_AT unit_NN1 rank_NN1 product_NN1 &lsqb;_( formula_NN1 &rsqb;_) ._. 
For_IF suppose_VV0 &lsqb;_( formula_NN1 &rsqb;_) ;_; then_RT &lsqb;_( formula_NN1 &rsqb;_) Postmultiply_RR this_DD1 by_II any_DD column_NN1 p_ZZ1 such_CS21 that_CS22 the_AT scalar_JJ product_NN1 &lsqb;_( formula_NN1 &rsqb;_) ;_; then_RT &lsqb;_( formula_NN1 &rsqb;_) Since_CS the_AT vector_NN1 x1_FO is_VBZ unique_JJ to_II a_AT1 scalar_JJ multiplier_NN1 ,_, we_PPIS2 may_VM identify_VVI &lsqb;_( formula_NN1 &rsqb;_) with_IW x1_FO and_CC similarly_RR &lsqb;_( formula_NN1 &rsqb;_) with_IW &lsqb;_( formula_NN1 &rsqb;_) ._. 
If_CS we_PPIS2 revert_VV0 to_II the_AT original_JJ problem_NN1 ,_, the_AT solution_NN1 of_IO Ax_NP1 =_FO &lsqb;_( formula_NN1 &rsqb;_) x_ZZ1 ,_, we_PPIS2 now_RT see_VV0 that_CST there_EX are_VBR just_RR n_ZZ1 eigenvalues_NN2 &lsqb;_( formula_NN1 &rsqb;_) s_ZZ1 and_CC that_CST correspondingly_RR there_EX are_VBR just_RR n_ZZ1 vectors_NN2 xs_MC2 ;_; i.e._REX we_PPIS2 have_VH0 n_ZZ1 equations_NN2 &lsqb;_( formula_NN1 &rsqb;_) We_PPIS2 can_VM combine_VVI them_PPHO2 all_DB into_II the_AT single_JJ equation_NN1 &lsqb;_( formula_NN1 &rsqb;_) or_CC more_RGR briefly_RR AX_NP1 =_FO X&lsqb;formula&rsqb;_VV0 where_RRQ X_ZZ1 is_VBZ ow_NN1 the_AT square_JJ matrix_NN1 made_VVD up_RP of_IO the_AT n_ZZ1 column_NN1 vectors_NN2 xs_MC2 ,_, and_CC &lsqb;_( formula_NN1 &rsqb;_) is_VBZ the_AT diagonal_JJ matrix_NN1 of_IO the_AT eigenvalues_NN2 ;_; and_CC our_APPGE problem_NN1 is_VBZ now_RT to_TO find_VVI the_AT matrices_NN2 X_ZZ1 and_CC &lsqb;_( formula_NN1 &rsqb;_) for_IF the_AT given_JJ A._NP1 We_PPIS2 may_VM not_XX one_PN1 further_RRR point_VVI ._. 
Let_VV0 d_ZZ1 be_VBI an_AT1 arbitrary_JJ diagonal_JJ matrix_NN1 ;_; then_RT it_PPH1 will_VM permute_VVI with_IW the_AT diagonal_JJ matrix_NN1 &lsqb;_( formula_NN1 &rsqb;_) Postmultiply_NP1 (_( 7_MC )_) by_II d_ZZ1 ;_; then_RT &lsqb;_( formula_NN1 &rsqb;_) showing_VVG that_CST Xd_NP1 is_VBZ a_AT1 solution_NN1 of_IO (_( 7_MC )_) ._. 
This_DD1 expresses_VVZ the_AT fact_NN1 that_CST the_AT columns_NN2 of_IO X_ZZ1 are_VBR each_DD1 arbitrary_JJ to_II a_AT1 scalar_JJ multiplier_NN1 ,_, since_CS in_II Xd_NP1 ,_, x1_FO is_VBZ multiplied_VVN by_II the_AT scalar_JJ d1_FO and_RR31 so_RR32 on_RR33 ._. 
If_CS (_( 7_MC )_) is_VBZ premultiplied_VVN by_II A_ZZ1 ,_, we_PPIS2 have_VH0 A(AX)_NN1 =_FO (_( AX_NP1 )_) &lsqb;_( formula_NN1 &rsqb;_) and_CC AX_NP1 is_VBZ apparently_RR an_AT1 alternative_JJ solution_NN1 for_IF X._NP1 But_CCB since_CS AX_NP1 =_FO Xgd_NP1 it_PPH1 is_VBZ X_ZZ1 postmultiplied_VVD by_II a_AT1 diagonal_JJ matrix_NN1 ,_, as_CSA before_RT ,_, except_CS21 that_CS22 here_RL the_AT diagonal_JJ matrix_NN1 is_VBZ not_XX arbitrary_JJ ._. 
The_AT matrix_NN1 equation_NN1 &lsqb;_( formula_NN1 &rsqb;_) arises_VVZ perhaps_RR most_RGT commonly_RR in_II the_AT study_NN1 of_IO the_AT natural_JJ frequencies_NN2 and_CC modes_NN2 of_IO vibration_NN1 of_IO an_AT1 undamped_JJ mechanical_JJ system_NN1 having_VHG n_ZZ1 degrees_NN2 of_IO freedom_NN1 ._. 
The_AT matrix_NN1 A_ZZ1 derives_VVZ from_II the_AT mechanical_JJ properties_NN2 (_( mass_NN1 and_CC stiffness_NN1 )_) of_IO the_AT system_NN1 ;_; &lsqb;_( formula_NN1 &rsqb;_) s_ZZ1 then_RT determines_VVZ the_AT square_NN1 of_IO a_AT1 natural_JJ frequency_NN1 ,_, and_CC xs_MC2 the_AT corresponding_JJ mode_NN1 of_IO distortion_NN1 ._. 
However_RR ,_, the_AT amplitude_NN1 is_VBZ not_XX determined_VVN ;_; xs_MC2 is_VBZ arbitrary_JJ to_II a_AT1 scalar_JJ multiplier_NN1 ._. 
In_II this_DD1 sense_NN1 an_AT1 arbitrary_JJ multiplier_NN1 ds_MC2 can_VM be_VBI regarded_VVN as_II an_AT1 amplitude_NN1 ._. 
Because_II21 of_II22 its_APPGE importance_NN1 in_II vibration_NN1 problems_NN2 ,_, the_AT matrix_NN1 X_ZZ1 ,_, which_DDQ is_VBZ the_AT array_NN1 of_IO the_AT individual_JJ eigenvectors_NN2 xs_MC2 ,_, is_VBZ usually_RR spoken_VVN of_IO as_CSA the_AT modal_JJ matrix_NN1 ,_, even_CS21 when_CS22 the_AT problem_NN1 from_II which_DDQ it_PPH1 derives_VVZ is_VBZ not_XX a_AT1 dynamical_JJ one_PN1 ._. 
Similarly_RR gd_JJ in_II vibration_NN1 problems_NN2 is_VBZ a_AT1 diagonal_JJ matrix_NN1 of_IO squares_NN2 of_IO natural_JJ frequencies_NN2 and_CC thus_RR defines_VVZ the_AT frequency_NN1 spectrum_NN1 of_IO the_AT system_NN1 ;_; accordingly_RR it_PPH1 is_VBZ referred_VVN to_II generally_RR as_CSA the_AT spectral_JJ matrix_NN1 ._. 
We_PPIS2 may_VM readily_RR show_VVI that_CST the_AT matrix_NN1 X_ZZ1 is_VBZ non-singular._JJ for_IF if_CS it_PPH1 is_VBZ not_XX ,_, let_VV0 us_PPIO2 first_MD suppose_VVI it_PPH1 to_TO be_VBI simply_RR degenerate_JJ ._. 
Then_RT there_EX will_VM be_VBI a_AT1 vector_NN1 p_ZZ1 ,_, unique_JJ apart_II21 from_II22 a_AT1 scalar_JJ factor_NN1 ,_, such_CS21 that_CS22 Xp_NP1 =_FO 0_MC ._. 
But_CCB in_II that_DD1 case_NN1 premultiplication_NN1 by_II A_ZZ1 gives_VVZ 0_MC =_FO AXp_NP1 =_FO X&lsqb;formula&rsqb;p_NN1 so_CS21 that_CS22 &lsqb;_( formula_NN1 &rsqb;_) p_ZZ1 provides_VVZ a_AT1 relation_NN1 different_JJ from_II p_ZZ1 between_II the_AT columns_NN2 of_IO X_ZZ1 ,_, which_DDQ is_VBZ impossible_JJ ._. 
Similarly_RR ,_, if_CS X_ZZ1 were_VBDR doubly_RR degenerate_JJ we_PPIS2 could_VM have_VHI only_RR two_MC vectors_NN2 p_ZZ1 satisfying_JJ Xp_NP1 =_FO 0_MC ,_, but_CCB each_DD1 of_IO these_DD2 would_VM give_VVI rise_NN1 to_II other_JJ different_JJ vectors_NN2 &lsqb;_( formula_NN1 &rsqb;_) p_ZZ1 ._. 
Hence_RR X_ZZ1 is_VBZ non-singular_JJ ._. 
We_PPIS2 may_VM accordingly_RR write_VVI (_( 7_MC )_) as_CSA &lsqb;_( formula_NN1 &rsqb;_) and_CC A_ZZ1 ,_, &lsqb;_( formula_NN1 &rsqb;_) are_VBR equivalent_JJ matrices_NN2 ._. 
If_CS &lsqb;_( formula_NN1 &rsqb;_) ,_, then_RT B_ZZ1 is_VBZ said_VVN to_TO be_VBI derived_VVN from_II A_ZZ1 by_II a_AT1 similar_JJ transformation_NN1 ._. 
This_DD1 type_NN1 of_IO transformation_NN1 is_VBZ of_IO great_JJ importance_NN1 ._. 
For_REX21 example_REX22 ,_, if_CS &lsqb;_( formula_NN1 &rsqb;_) then_RT &lsqb;_( formula_NN1 &rsqb;_) where_RRQ &lsqb;_( formula_NN1 &rsqb;_) is_VBZ the_AT diagonal_JJ matrix_NN1 of_IO the_AT squares_NN2 of_IO the_AT eigenvalues_NN2 ._. 
Similarly_RR &lsqb;_( formula_NN1 &rsqb;_) and_CC more_RGR generally_RR &lsqb;_( formula_NN1 &rsqb;_) where_RRQ P(A)_NN1 is_VBZ any_DD polynomial_NN1 in_II the_AT matrix_NN1 A._NP1 We_PPIS2 now_RT revert_VV0 briefly_RR to_II Equations_NN2 (_( 2_MC )_) and_CC (_( 3_MC )_) in_BCL21 order_BCL22 to_TO define_VVI the_AT companion_NN1 matrix_NN1 C_ZZ1 of_IO &lsqb;_( formula_NN1 &rsqb;_) (_( &lsqb;_( formula_NN1 &rsqb;_) )_) ._. 
This_DD1 is_VBZ &lsqb;_( formula_NN1 &rsqb;_) Condensation_NN1 of_IO &lsqb;_( formula_NN1 &rsqb;_) shows_VVZ that_CST &lsqb;_( formula_NN1 &rsqb;_) so_CS21 that_CS22 the_AT eigenvalues_NN2 of_IO A_ZZ1 and_CC C_ZZ1 are_VBR the_AT same_DA ._. 
We_PPIS2 shall_VM give_VVI an_AT1 example_NN1 of_IO the_AT utility_NN1 of_IO (_( 10_MC )_) later_JJR (_( 2.9_MC )_) ._. 
1.17_MC THE_AT CAYLEY-HAMILTON_NP1 THEOREM_NN1 This_DD1 important_JJ theorem_NN1 states_VVZ that_CST any_DD square_JJ matrix_NN1 A_ZZ1 satisfies_VVZ its_APPGE own_DA characteristic_JJ equation_NN1 ._. 
First_MD ,_, suppose_VV0 as_CSA above_RL that_CST the_AT eigenvalues_NN2 are_VBR all_DB distinct_JJ ;_; then_RT in_II31 view_II32 of_II33 (_( 1.16.9_MC )_) we_PPIS2 may_VM write_VVI the_AT polynomial_NN1 &lsqb;_( formula_NN1 &rsqb;_) (_( A_ZZ1 )_) as_CSA &lsqb;_( formula_NN1 &rsqb;_) Now_RT the_AT diagonal_JJ matrix_NN1 &lsqb;_( formula_NN1 &rsqb;_) has_VHZ for_IF diagonal_JJ elements_NN2 &lsqb;_( formula_NN1 &rsqb;_) all_DB of_IO which_DDQ vanish_VV0 ;_; i.e._REX &lsqb;_( formula_NN1 &rsqb;_) is_VBZ null_JJ ._. 
Hence_RR &lsqb;_( formula_NN1 &rsqb;_) (_( A_ZZ1 )_) =_FO 0_MC ,_, which_DDQ is_VBZ the_AT Cayley-Hamilton_NP1 theorem_NN1 ._. 
The_AT equation_NN1 is_VBZ true_JJ even_CS21 when_CS22 the_AT eigenvalues_NN2 are_VBR not_XX all_DB different_JJ ._. 
For_CS we_PPIS2 know_VV0 that_CST in_RR21 general_RR22 &lsqb;_( formula_NN1 &rsqb;_) where_RRQ &lsqb;_( formula_NN1 &rsqb;_) is_VBZ the_AT adjugate_NN1 of_IO (_( &lsqb;_( formula_NN1 &rsqb;_) I_ZZ1 A_ZZ1 )_) ._. 
Since_CS both_RR &lsqb;_( formula_NN1 &rsqb;_) and_CC gd_JJ (_( &lsqb;_( formula_NN1 &rsqb;_) )_) I_PPIS1 are_VBR lambda-matrices_NN2 ,_, we_PPIS2 amy_NN1 write_VV0 them_PPHO2 as_CSA polynomials_NN2 in_II &lsqb;_( formula_NN1 &rsqb;_) with_IW matrix_NN1 coefficients_NN2 ;_; let_VV0 us_PPIO2 write_VVI the_AT equation_NN1 as_CSA &lsqb;_( formula_NN1 &rsqb;_) Then_RT on_II multiplying_VVG out_RP and_CC identifying_VVG coefficients_NN2 of_IO &lsqb;_( formula_NN1 &rsqb;_) ,_, we_PPIS2 have_VH0 &lsqb;_( formula_NN1 &rsqb;_) Premultiply_RR the_AT first_MD of_IO these_DD2 relations_NN2 by_II &lsqb;_( formula_NN1 &rsqb;_) ,_, the_AT second_NNT1 by_II &lsqb;_( formula_NN1 &rsqb;_) ,_, etc._RA and_CC add_VV0 :_: the_AT result_NN1 is_VBZ &lsqb;_( formula_NN1 &rsqb;_) (_( A_ZZ1 )_) =_FO 0_MC ._. 
Equations_NN2 (_( 1.16.3_MC )_) and_CC (_( 1.16.4_MC )_) show_VV0 that_CST the_AT characteristic_JJ equation_NN1 may_VM be_VBI written_VVN in_II the_AT alternative_JJ forms_NN2 &lsqb;_( formula_NN1 &rsqb;_) It_PPH1 follows_VVZ that_CST the_AT Cayley-Hamilton_NP1 theorem_NN1 &lsqb;_( formula_NN1 &rsqb;_) can_VM be_VBI written_VVN in_II the_AT alternative_JJ form_NN1 ,_, in_II which_DDQ for_IF consistency_NN1 each_DD1 factor_NN1 has_VHZ been_VBN multiplied_VVN by_II -1_MC ,_, &lsqb;_( formula_NN1 &rsqb;_) It_PPH1 will_VM be_VBI observed_VVN that_CST ,_, with_IW eigenvalues_NN2 all_DB different_JJ ,_, each_DD1 of_IO these_DD2 matrices_NN2 has_VHZ degeneracy_NN1 1_MC1 ,_, the_AT product_NN1 of_IO all_DB n_ZZ1 factors_NN2 having_VHG degeneracy_NN1 n_ZZ1 :_: i.e._REX it_PPH1 is_VBZ null_JJ ;_; this_DD1 is_VBZ an_AT1 illustration_NN1 of_IO Sylvester_NP1 's_GE law_NN1 of_IO degeneracy_NN1 (_( Theorem_NN1 XII_MC of_IO 1.22_MC )_) ._. 
Since_CS &lsqb;_( formula_NN1 &rsqb;_) (_( A_ZZ1 )_) vanishes_VVZ ,_, so_RG also_RR do_VD0 &lsqb;_( formula_NN1 &rsqb;_) ,_, etc._RA it_PPH1 follows_VVZ that_CST any_DD power_NN1 n_ZZ1 or_CC more_DAR of_IO A_ZZ1 ,_, or_CC any_DD polynomial_NN1 degree_NN1 of_IO n_ZZ1 or_CC more_RRR in_II A_ZZ1 ,_, can_VM be_VBI expressed_VVN as_II a_AT1 polynomial_NN1 in_II A_ZZ1 of_IO degree_NN1 n_ZZ1 1_MC1 ._. 
Furthermore_RR ,_, if_CS A_ZZ1 is_VBZ non-singular_JJ ,_, &lsqb;_( formula_NN1 &rsqb;_) may_VM be_VBI expressed_VVN as_II a_AT1 polynomial_NN1 of_IO degree_NN1 n_ZZ1 1_MC1 in_II A_ZZ1 ,_, and_CC therefore_RR also_RR &lsqb;_( formula_NN1 &rsqb;_) and_CC higher_JJR negative_JJ powers_NN2 ._. 
Examples_NN2 of_IO the_AT utility_NN1 of_IO this_DD1 theorem_NN1 will_VM appear_VVI later._NNU 1.18_MC SYLVESTER_NP1 'S_GE EXPANSION_NN1 THEOREM_NN1 We_PPIS2 revert_VV0 to_II Equation_NN1 (_( 1.16.9_MC )_) and_CC write_VV0 &lsqb;_( formula_NN1 &rsqb;_) ,_, to_TO conform_VVI with_IW the_AT notation_NN1 for_IF the_AT adjugate_NN1 of_IO X._NP1 Then_RT &lsqb;_( formula_NN1 &rsqb;_) Now_RT P(A)_NN1 is_VBZ the_AT diagonal_JJ matrix_NN1 of_IO which_DDQ the_AT diagonal_JJ elements_NN2 are_VBR &lsqb;_( formula_NN1 &rsqb;_) in_II succession_NN1 ._. 
Once_RR21 again_RR22 we_PPIS2 here_RL assume_VV0 the_AT eigenvalues_NN2 &lsqb;_( formula_NN1 &rsqb;_) s_ZZ1 to_TO be_VBI all_RR different_JJ ._. 
We_PPIS2 now_RT write_VV0 &lsqb;_( formula_NN1 &rsqb;_) so_CS21 that_CS22 each_DD1 square_JJ matrix_NN1 is_VBZ null_JJ except_II21 for_II22 a_AT1 unit_NN1 in_II the_AT appropriate_JJ diagonal_JJ position_NN1 ._. 
In_II31 terms_II32 of_II33 the_AT isolating_JJ vector_NN1 ei_NNU this_DD1 may_VM be_VBI written_VVN &lsqb;_( formula_NN1 &rsqb;_) and_CC if_CS P(A)_NN1 is_VBZ expanded_VVN in_II this_DD1 way_NN1 ,_, the_AT first_MD term_NN1 ,_, which_DDQ is_VBZ typical_JJ will_VM be_VBI &lsqb;_( formula_NN1 &rsqb;_) The_AT square_JJ matrices_NN2 on_II the_AT right_NN1 of_IO these_DD2 equations_NN2 are_VBR all_DB of_IO unit_NN1 rank_NN1 ;_; we_PPIS2 may_VM write_VVI the_AT last_MD as_CSA &lsqb;_( formula_NN1 &rsqb;_) where_RRQ x1_FO is_VBZ the_AT first_MD column_NN1 of_IO X_ZZ1 ,_, and_CC &lsqb;_( formula_NN1 &rsqb;_) the_AT first_MD row_NN1 of_IO &lsqb;_( formula_NN1 &rsqb;_) ._. 
We_PPIS2 thus_RR establish_VV0 Sylvester_NP1 's_GE theorem_NN1 for_IF the_AT expansion_NN1 of_IO P(A)_NN1 ,_, viz._REX &lsqb;_( formula_NN1 &rsqb;_) where_RRQ &lsqb;_( formula_NN1 &rsqb;_) The_AT unit_NN1 rank_NN1 matrices_NN2 &lsqb;_( formula_NN1 &rsqb;_) which_DDQ are_VBR known_VVN as_II the_AT constituent_NN1 matrices_NN2 of_IO A_ZZ1 ,_, have_VH0 interesting_JJ and_CC important_JJ properties_NN2 ._. 
Thus_RR :_: (_( i_ZZ1 )_) As_CSA has_VHZ been_VBN noted_VVN earlier_RRR ,_, in_II 1.16_MC &lsqb;_( formula_NN1 &rsqb;_) is_VBZ proportional_JJ to_II &lsqb;_( formula_NN1 &rsqb;_) ,_, where_RRQ &lsqb;_( formula_NN1 &rsqb;_) is_VBZ the_AT adjugate_NN1 of_IO (_( &lsqb;_( formula_NN1 &rsqb;_) I_ZZ1 A_ZZ1 )_) ._. 
(_( ii_MC )_) We_PPIS2 defined_VVD &lsqb;_( formula_NN1 &rsqb;_) as_CSA &lsqb;_( formula_NN1 &rsqb;_) ,_, hence_RR &lsqb;_( formula_NN1 &rsqb;_) This_DD1 equation_NN1 requires_VVZ ,_, for_IF all_DB the_AT rows_NN2 &lsqb;_( formula_NN1 &rsqb;_) and_CC columns_NN2 &lsqb;_( formula_NN1 &rsqb;_) &lsqb;_( formula_NN1 &rsqb;_) Hence_RR &lsqb;_( formula_NN1 &rsqb;_) since_CS the_AT inner_JJ product_NN1 (_( scalar_JJ )_) is_VBZ unit_NN1 ._. 
Hence_RR in_RR21 general_RR22 &lsqb;_( formula_NN1 &rsqb;_) On_II31 account_II32 of_II33 this_DD1 property_NN1 ,_, &lsqb;_( formula_NN1 &rsqb;_) is_VBZ said_VVN to_TO be_VBI idempotent._NNU (_( iii_MC )_) We_PPIS2 have_VH0 also_RR &lsqb;_( formula_NN1 &rsqb;_) since_CS the_AT inner_JJ product_NN1 vanishes._NNU (_( iv_MC )_) If_CS in_II Sylvester_NP1 's_GE expansion_NN1 we_PPIS2 put_VV0 P(A)_NN1 ,_, which_DDQ is_VBZ a_AT1 arbitrary_JJ polynomial_NN1 ,_, equal_JJ to_II I_ZZ1 ,_, so_CS21 that_CS22 &lsqb;_( formula_NN1 &rsqb;_) ,_, then_RT &lsqb;_( formula_NN1 &rsqb;_) As_II a_AT1 simple_JJ example_NN1 of_IO the_AT consistency_NN1 of_IO the_AT theorem_NN1 ,_, let_VV0 us_PPIO2 write_VVI Sylvester_NP1 's_GE expansion_NN1 for_IF A_ZZ1 as_CSA &lsqb;_( formula_NN1 &rsqb;_) Let_VV0 us_PPIO2 square_RR both_DB2 sides_NN2 ;_; then_RT &lsqb;_( formula_NN1 &rsqb;_) and_CC on_II31 account_II32 of_II33 the_AT properties_NN2 above_RL &lsqb;_( formula_NN1 &rsqb;_) in_II31 accordance_II32 with_II33 Sylvester_NP1 's_GE theorem._NNU 1.19_MC ORTHOGONALITY_NP1 AND_CC BI-ORTHOGONALITY_NN1 OF_IO THE_AT MODAL_JJ MATRIX_NN1 We_PPIS2 consider_VV0 first_MD the_AT eigenvalues_NN2 (_( supposed_VVN all_DB different_JJ )_) and_CC modal_JJ matrix_NN1 of_IO a_AT1 symmetric_JJ matrix_NN1 A._NP1 then_RT by_II (_( 1.16.7_MC )_) AX_NP1 =_FO X&lsqb;formula&rsqb;_NP1 and_CC on_II premultiplication_NN1 &lsqb;_( formula_NN1 &rsqb;_) Transpose_VV0 this_DD1 ;_; then_RT since_CS &lsqb;_( formula_NN1 &rsqb;_) we_PPIS2 obtain_VV0 &lsqb;_( formula_NN1 &rsqb;_) Comparison_NN1 of_IO (_( 1_MC1 )_) and_CC (_( 2_MC )_) shows_VVZ that_CST &lsqb;_( formula_NN1 &rsqb;_) so_CS21 that_CS22 &lsqb;_( formula_NN1 &rsqb;_) permutes_VVZ with_IW &lsqb;_( formula_NN1 &rsqb;_) ,_, a_AT1 diagonal_JJ matrix_NN1 with_IW all_DB its_APPGE diagonal_JJ elements_NN2 different_JJ ._. 
Hence_RR (_( see_VV0 1.9_MC )_) &lsqb;_( formula_NN1 &rsqb;_) must_VM also_RR be_VBI diagonal_JJ and_CC it_PPH1 then_RT follows_VVZ that_CST &lsqb;_( formula_NN1 &rsqb;_) is_VBZ also_RR diagonal_JJ ._. 
Now_RT we_PPIS2 have_VH0 seen_VVN that_CST we_PPIS2 can_VM use_VVI Xd_NP1 in_II31 place_II32 of_II33 X_ZZ1 ,_, where_CS d_ZZ1 is_VBZ an_AT1 arbitrary_JJ diagonal_JJ matrix_NN1 ;_; and_CC it_PPH1 therefore_RR follows_VVZ that_CST &lsqb;_( formula_NN1 &rsqb;_) is_VBZ also_RR diagonal_JJ ._. 
Now_RT the_AT ith_MD diagonal_JJ element_NN1 of_IO this_DD1 product_NN1 is_VBZ &lsqb;_( formula_NN1 &rsqb;_) and_CC we_PPIS2 may_VM clearly_RR choose_VVI di_FW to_TO make_VVI this_DD1 element_NN1 unity_NN1 ._. 
If_CS this_DD1 is_VBZ done_VDN for_IF all_DB i_ZZ1 ,_, and_CC we_PPIS2 write_VV0 Xn_FO for_IF Xd_NP1 ,_, then_RT &lsqb;_( formula_NN1 &rsqb;_) so_CS21 that_CS22 the_AT modal_JJ matrix_NN1 Xn_FO ,_, chosen_VVN in_II this_DD1 way_NN1 ,_, is_VBZ orthogonal_JJ and_CC we_PPIS2 say_VV0 that_CST Xn_FO is_VBZ the_AT normalised_JJ for_CS of_IO the_AT modal_JJ matrix_NN1 ._. 
Next_MD consider_VV0 the_AT more_RGR general_JJ case_NN1 when_CS A_ZZ1 is_VBZ not_XX symmetric_JJ ,_, but_CCB is_VBZ expressed_VVN as_II the_AT product_NN1 of_IO two_MC symmetric_JJ factors_NN2 ._. 
Such_DA factorisation_NN1 is_VBZ always_RR possible_JJ ,_, in_II a_AT1 number_NN1 of_IO ways_NN2 ;_; for_REX21 example_REX22 ,_, since_CS AX_NP1 =_FO XA_NP1 then_RT &lsqb;_( formula_NN1 &rsqb;_) so_CS21 that_CS22 A_ZZ1 has_VHZ been_VBN factorised_VVN into_II two_MC symmetric_JJ matrices_NN2 ;_; however_RR ,_, this_DD1 particular_JJ factorisation_NN1 presupposes_VVZ a_AT1 knowledge_NN1 of_IO X_ZZ1 and_CC &lsqb;_( formula_NN1 &rsqb;_) ._. 
Suppose_VV0 instead_RR that_CST we_PPIS2 can_VM write_VVI A_ZZ1 as_CSA &lsqb;_( formula_NN1 &rsqb;_) where_RRQ B_ZZ1 ,_, C_ZZ1 are_VBR symmetric_JJ ;_; then_RT we_PPIS2 have_VH0 &lsqb;_( formula_NN1 &rsqb;_) or_CC BX_NP1 =_FO CX&lsqb;formula&rsqb;_NP1 ._. 
Dynamical_JJ problems_NN2 frequently_RR arise_VV0 in_II this_DD1 form_NN1 directly_RR ._. 
Now_RT as_CSA before_RT ,_, if_CS we_PPIS2 premultiply_RR by_II &lsqb;_( formula_NN1 &rsqb;_) and_CC transpose_VV0 ,_, since_CS &lsqb;_( formula_NN1 &rsqb;_) and_CC &lsqb;_( formula_NN1 &rsqb;_) &lsqb;_( formula_NN1 &rsqb;_) so_CS21 that_CS22 &lsqb;_( formula_NN1 &rsqb;_) permutes_VVZ with_IW &lsqb;_( formula_NN1 &rsqb;_) and_CC is_VBZ thus_RR diagonal_JJ ,_, whence_RRQ it_PPH1 follows_VVZ that_CST &lsqb;_( formula_NN1 &rsqb;_) is_VBZ diagonal_JJ also_RR ._. 
Clearly_RR we_PPIS2 can_VM not_XX normalise_VVI X_ZZ1 in_II this_DD1 case_NN1 to_TO make_VVI both_RR &lsqb;_( formula_NN1 &rsqb;_) and_CC &lsqb;_( formula_NN1 &rsqb;_) unit_NN1 matrices_NN2 ;_; we_PPIS2 say_VV0 therefore_RR that_CST in_II this_DD1 case_NN1 X_ZZ1 is_VBZ bi-orthogonal_JJ or_CC has_VHZ a_AT1 generalised_JJ orthogonal_JJ form_NN1 ._. 
We_PPIS2 may_VM pursue_VVI the_AT matter_NN1 a_RR21 little_RR22 further_RRR ,_, however_RR ._. 
If_CS as_CSA previously_RR we_PPIS2 write_VV0 Xd_NP1 for_IF X_ZZ1 ,_, where_CS d_ZZ1 is_VBZ an_AT1 arbitrary_JJ diagonal_JJ matrix_NN1 ,_, then_RT we_PPIS2 obtain_VV0 &lsqb;_( formula_NN1 &rsqb;_) as_CSA diagonal_JJ matrices_NN2 ._. 
Now_RT the_AT ith_MD diagonal_JJ element_NN1 of_IO &lsqb;_( formula_NN1 &rsqb;_) &lsqb;_( formula_NN1 &rsqb;_) where_RRQ the_AT bracketed_JJ term_NN1 is_VBZ the_AT quadratic_JJ form_NN1 obtained_VVN from_II C_NP1 with_IW the_AT ith_MD eigenvector_NN1 xi_NN1 ._. 
This_DD1 quadratic_JJ form_NN1 is_VBZ not_XX ,_, as_CSA previously_RR ,_, a_AT1 sum_NN1 of_IO squares_NN2 ;_; but_CCB provided_CS it_PPH1 is_VBZ not_XX zero_VV0 we_PPIS2 could_VM choose_VVI di_FW to_TO make_VVI the_AT element_NN1 unity_NN1 ,_, when_CS we_PPIS2 should_VM have_VHI (_( with_IW X_ZZ1 for_IF Xd_NP1 )_) &lsqb;_( formula_NN1 &rsqb;_) On_II31 account_II32 of_II33 the_AT properties_NN2 established_VVN here_RL ,_, it_PPH1 is_VBZ usual_JJ to_TO say_VVI that_CST the_AT modes_NN2 of_IO a_AT1 system_NN1 are_VBR orthogonal_JJ to_II each_PPX221 other._PPX222 1.20_MC CANONICAL_JJ FORMS_NN2 OBTAINED_VVN BY_II SIMILAR_JJ TRANSFORMATION_NN1 We_PPIS2 consider_VV0 a_AT1 matrix_NN1 A_ZZ1 ,_, which_DDQ for_IF the_AT moment_NN1 we_PPIS2 regard_VV0 as_II having_VHG distinct_JJ eigenvalues_NN2 ._. 
We_PPIS2 have_VH0 seen_VVN (_( 1.12_MC )_) that_CST by_II elementary_JJ operations_NN2 we_PPIS2 can_VM perform_VVI an_AT1 equivalent_JJ transformation_NN1 C_ZZ1 =_FO PAQ_NP1 and_CC C_ZZ1 has_VHZ a_AT1 special_JJ canonical_JJ form_NN1 ,_, either_RR the_AT unit_NN1 matrix_NN1 or_CC containing_VVG a_AT1 diagonal_JJ unit_NN1 submatrix_NN1 ._. 
Let_VV0 us_PPIO2 now_RT impose_VVI the_AT restriction_NN1 that_CST whenever_RRQV we_PPIS2 perform_VV0 an_AT1 elementary_JJ postmultiplying_JJ operation_NN1 to_TO make_VVI up_RP Q_ZZ1 ,_, we_PPIS2 premultiply_RR by_II the_AT inverse_JJ operation_NN1 to_TO make_VVI up_RP P_ZZ1 ,_, so_CS21 that_CS22 &lsqb;_( formula_NN1 &rsqb;_) ._. 
In_II this_DD1 case_NN1 ,_, we_PPIS2 can_VM not_XX reduce_VVI C_NP1 to_II a_AT1 diagonal_JJ matrix_NN1 of_IO units_NN2 ,_, since_CS evidently_RR &lsqb;_( formula_NN1 &rsqb;_) ._. 
However_RR ,_, a_AT1 simple_JJ and_CC powerful_JJ canonical_JJ form_NN1 is_VBZ obtainable_JJ ._. 
The_AT matrix_NN1 A_ZZ1 has_VHZ the_AT characteristic_JJ matrix_NN1 (_( &lsqb;_( formula_NN1 &rsqb;_) I_ZZ1 A_ZZ1 )_) ._. 
If_CS this_DD1 is_VBZ premultiplied_VVN by_II &lsqb;_( formula_NN1 &rsqb;_) and_CC postmultiplied_VVD by_II Q_ZZ1 ,_, then_RT &lsqb;_( formula_NN1 &rsqb;_) so_CS21 that_CS22 A_ZZ1 and_CC &lsqb;_( formula_NN1 &rsqb;_) have_VH0 the_AT same_DA eigenvalues_NN2 ._. 
In_RR21 particular_RR22 ,_, if_CS we_PPIS2 identify_VV0 Q_ZZ1 with_IW the_AT modal_JJ matrix_NN1 X_ZZ1 ,_, then_RT &lsqb;_( formula_NN1 &rsqb;_) by_II (_( 1.16.8_MC )_) ._. 
Hence_RR ,_, as_CSA is_VBZ otherwise_RR obvious_JJ from_II the_AT derivation_NN1 of_IO (_( 1.16.8_MC )_) ,_, the_AT eigenvalues_NN2 of_IO A_ZZ1 are_VBR given_VVN directly_RR by_II &lsqb;_( formula_NN1 &rsqb;_) ,_, etc_RA ._. 
In_II this_DD1 case_NN1 ,_, therefore_RR ,_, we_PPIS2 may_VM regard_VVI &lsqb;_( formula_NN1 &rsqb;_) as_II the_AT simple_JJ canonical_JJ form_NN1 for_IF a_AT1 matrix_NN1 having_VHG distinct_JJ eigenvalues_NN2 ._. 
Note_VV0 that_CST if_CS &lsqb;_( formula_NN1 &rsqb;_) and_CC &lsqb;_( formula_NN1 &rsqb;_) then_RT &lsqb;_( formula_NN1 &rsqb;_) so_CS21 that_CS22 ,_, if_CS X_ZZ1 is_VBZ the_AT modal_JJ matrix_NN1 of_IO A_ZZ1 ,_, QX_NP1 is_VBZ the_AT modal_JJ matrix_NN1 of_IO B._NP1 The_AT great_JJ majority_NN1 of_IO mechanical_JJ problems_NN2 give_VV0 rise_NN1 to_II matrices_NN2 having_VHG distinct_JJ eigenvalues_NN2 ;_; however_RR ,_, problems_NN2 involving_VVG equal_JJ eigenvalues_NN2 are_VBR by_RR31 no_RR32 means_RR33 unknown_JJ ._. 
For_REX21 example_REX22 ,_, a_AT1 rigid_JJ body_NN1 may_VM have_VHI some_DD of_IO its_APPGE motions_NN2 constrained_VVN by_II springs_NN2 ,_, but_CCB two_MC (_( or_CC more_RRR )_) unconstrained_JJ ;_; it_PPH1 will_VM then_RT have_VHI two_MC (_( or_CC more_RRR )_) equal_JJ zero_NN1 frequencies_NN2 ._. 
The_AT canonical_JJ form_NN1 obtained_VVN by_II a_AT1 similar_JJ transformation_NN1 of_IO a_AT1 matrix_NN1 having_VHG equal_JJ eigenvalues_NN2 ,_, when_CS reduced_VVN to_II its_APPGE simplest_JJT form_NN1 ,_, has_VHZ the_AT eigenvalues_NN2 in_II the_AT principal_JJ diagonal_JJ ,_, with_IW equal_JJ eigenvalues_NN2 in_II groups_NN2 ;_; elsewhere_RL the_AT matrix_NN1 is_VBZ null_JJ except_CS21 that_CS22 there_EX may_VM be_VBI units_NN2 in_II the_AT superdiagonal_JJ ._. 
Thus_RR ,_, for_REX21 example_REX22 ,_, a_AT1 matrix_NN1 of_IO order_NN1 6_MC having_VHG three_MC equal_JJ eigenvalues_NN2 ga_NN1 ,_, two_MC equal_JJ eigenvalues_NN2 gb_NNU ,_, and_CC an_AT1 unrepeated_JJ eigenvalue_NN1 gg_NNU ,_, would_VM have_VHI as_II its_APPGE canonical_JJ form_NN1 &lsqb;_( formula_NN1 &rsqb;_) where_RRQ each_DD1 x_ZZ1 may_VM be_VBI zero_MC or_CC unity_NN1 ;_; it_PPH1 occurs_VVZ only_RR within_II the_AT submatrices_NN2 containing_VVG groups_NN2 of_IO equal_JJ eigenvalues_NN2 ._. 
C_ZZ1 is_VBZ sometimes_RT called_VVN the_AT Jordan_NP1 canonical_JJ form_NN1 and_CC a_AT1 submatrix_NN1 such_II21 as_II22 &lsqb;_( formula_NN1 &rsqb;_) is_VBZ known_VVN as_II a_AT1 Jordan_NP1 block_NN1 ._. 
If_CS in_II C_NP1 a_AT1 unit_NN1 occurs_VVZ at_II a_AT1 position_NN1 x_ZZ1 ,_, it_PPH1 can_VM not_XX be_VBI removed_VVN by_II a_AT1 similar_JJ transformation_NN1 ._. 
This_DD1 is_VBZ readily_RR proved_VVN as_CSA follows_VVZ ._. 
Suppose_VV0 we_PPIS2 wish_VV0 to_TO remove_VVI the_AT unit_NN1 from_II the_AT superdiagonal_JJ in_II the_AT matrix_NN1 &lsqb;_( formula_NN1 &rsqb;_) by_II31 means_II32 of_II33 a_AT1 similar_JJ transformation_NN1 ;_; we_PPIS2 can_VM write_VVI the_AT most_RGT general_JJ transformation_NN1 matrix_NN1 as_CSA &lsqb;_( formula_NN1 &rsqb;_) Then_RT &lsqb;_( formula_NN1 &rsqb;_) and_CC if_CS both_DB2 off-diagonal_JJ elements_NN2 are_VBR to_TO vanish_VVI ,_, then_RT &lsqb;_( formula_NN1 &rsqb;_) Provided_VVD &lsqb;_( formula_NN1 &rsqb;_) we_PPIS2 may_VM satisfy_VVI (_( 4_MC )_) and_CC &lsqb;_( formula_NN1 &rsqb;_) :_: for_REX21 example_REX22 ,_, if_CS we_PPIS2 choose_VV0 &lsqb;_( formula_NN1 &rsqb;_) ._. 
Then_RT &lsqb;_( formula_NN1 &rsqb;_) However_RR ,_, if_CS &lsqb;_( formula_NN1 &rsqb;_) then_RT the_AT only_JJ solution_NN1 of(4)_FO is_VBZ c_ZZ1 =_FO d_ZZ1 =_FO 0_MC when_RRQ ad_NN1 bc_RA vanishes_VVZ ,_, so_CS21 that_CS22 &lsqb;_( formula_NN1 &rsqb;_) does_VDZ not_XX exist_VVI ._. 
Hence_RR ,_, if_CS a_AT1 unit_NN1 exists_VVZ (_( it_PPH1 may_VM not_XX )_) in_II the_AT superdiagonal_JJ element_NN1 adjacent_II21 to_II22 two_MC equal_JJ eigenvalues_NN2 ,_, it_PPH1 can_VM not_XX be_VBI removed_VVN by_II a_AT1 similar_JJ transformation_NN1 ._. 
The_AT reason_NN1 for_IF the_AT existence_NN1 or_CC absence_NN1 of_IO a_AT1 unit_NN1 in_II the_AT superdiagonal_JJ is_VBZ easy_JJ to_TO see_VVI ._. 
Suppose_VV0 C_ZZ1 in_II (_( 3_MC )_) is_VBZ derived_VVN by_II a_AT1 similar_JJ transformation_NN1 from_II a_AT1 matrix_NN1 A_ZZ1 ,_, to_II which_DDQ it_PPH1 is_VBZ therefore_RR equivalent_JJ ._. 
Then_RT if_CS &lsqb;_( formula_NN1 &rsqb;_) I-A_ZZ1 is_VBZ simply_RR degenerate_JJ when_CS &lsqb;_( formula_NN1 &rsqb;_) =_FO ga_NN1 so_RG also_RR is_VBZ &lsqb;_( formula_NN1 &rsqb;_) I_ZZ1 C._ZZ1 There_EX must_VM in_II consequence_NN1 be_VBI two_MC units_NN2 in_II the_AT top_NN1 two_MC places_NN2 of_IO the_AT superdiagonal_JJ ,_, for_IF without_IW them_PPHO2 &lsqb;_( formula_NN1 &rsqb;_) I_ZZ1 C_ZZ1 would_VM have_VHI three_MC null_JJ rows_NN2 and_CC three_MC null_JJ columns_NN2 ,_, and_CC so_RR be_VBI triply_RR degenerate_JJ ._. 
With_IW two_MC units_NN2 present_VV0 ,_, only_RR the_AT first_MD column_NN1 and_CC third_MD row_NN1 are_VBR null_JJ ,_, so_CS21 that_CS22 &lsqb;_( formula_NN1 &rsqb;_) I_ZZ1 C_ZZ1 is_VBZ simply_RR degenerate_JJ as_CSA required_VVN ._. 
Evidently_RR the_AT introduction_NN1 of_IO a_AT1 unit_NN1 reduces_VVZ the_AT degeneracy_NN1 of_IO &lsqb;_( formula_NN1 &rsqb;_) I_ZZ1 C_ZZ1 by_II unity_NN1 ._. 
It_PPH1 is_VBZ clear_JJ from_II the_AT derivation_NN1 of_IO the_AT canonical_JJ form_NN1 that_CST it_PPH1 is_VBZ conventional_JJ ,_, but_CCB not_XX necessary_JJ ,_, to_TO put_VVI equal_JJ eigenvalues_NN2 in_II groups_NN2 ,_, and_CC to_TO put_VVI units_NN2 in_II the_AT superdiagonal_JJ ._. 
In_II a_AT1 practical_JJ case_NN1 ,_, we_PPIS2 might_VM find_VVI an_AT1 intermediate_JJ form_NN1 for_IF the_AT matrix_NN1 C_ZZ1 of_IO (_( 3_MC )_) to_TO be_VBI (_( we_PPIS2 shall_VM discuss_VVI the_AT meaning_NN1 of_IO x_ZZ1 and_CC y_ZZ1 )_) &lsqb;_( formula_NN1 &rsqb;_) Now_RT suppose_VV0 &lsqb;_( formula_NN1 &rsqb;_) ,_, and_CC therefore_RR &lsqb;_( formula_NN1 &rsqb;_) to_TO be_VBI simply_RR degenerate_JJ when_CS &lsqb;_( formula_NN1 &rsqb;_) Then_RT &lsqb;_( formula_NN1 &rsqb;_) must_VM also_RR be_VBI simply_RR degenerate_JJ ._. 
This_DD1 requires_VVZ that_CST either_RR x_ZZ1 or_CC y_ZZ1 is_VBZ nonzero_NN1 ,_, the_AT other_NN1 being_VBG zero_MC ;_; only_RR so_RR does_VDZ the_AT matrix_NN1 have_VHI only_RR one_MC1 null_JJ column_NN1 and_CC row_NN1 ._. 
Suppose_VV0 x_ZZ1 is_VBZ zero_MC ,_, so_CS21 that_CS22 the_AT second_MD row_NN1 and_CC fifth_MD column_NN1 are_VBR null_JJ ._. 
Then_RT y_ZZ1 must_VM not_XX vanish_VVI ._. 
Now_RT ,_, by_II further_JJR operations_NN2 of_IO type_NN1 I_ZZ1 (_( i_ZZ1 ,_, j_ZZ1 )_) we_PPIS2 can_VM bring_VVI the_AT two_MC roots_NN2 into_II adjacent_JJ places_NN2 in_II the_AT diagonal_JJ ,_, when_CS -y_ZZ1 will_VM move_VVI to_II the_AT infradiagonal_JJ and_CC a_AT1 zero_NN1 ,_, replacing_VVG x_ZZ1 ,_, will_VM appear_VVI in_II the_AT superdiagonal_JJ ._. 
A_AT1 similar_JJ transformation_NN1 of_IO the_AT submatrix_NN1 then_RT gives_VVZ &lsqb;_( formula_NN1 &rsqb;_) so_CS21 that_CS22 ,_, so_RG far_RR as_CSA &lsqb;_( formula_NN1 &rsqb;_) is_VBZ concerned_JJ ,_, we_PPIS2 have_VH0 arrived_VVN at_II the_AT conventional_JJ canonical_JJ form_NN1 ._. 
The_AT conventional_JJ form_NN1 is_VBZ generally_RR convenient_JJ ;_; it_PPH1 is_VBZ useful_JJ to_TO have_VHI equal_JJ eigenvalues_NN2 in_II groups_NN2 and_CC to_II the_AT use_NN1 superdiagonal_JJ ;_; but_CCB in_RR21 particular_RR22 cases_VVZ it_PPH1 may_VM be_VBI well_JJ to_TO use_VVI a_AT1 nonzero_NN1 quantity_NN1 other_II21 than_II22 a_AT1 unit_NN1 in_II the_AT superdiagonal_JJ ._. 
For_REX21 example_REX22 ,_, if_CS the_AT eigenvalues_NN2 are_VBR numerically_RR small_JJ ,_, a_AT1 unit_NN1 could_VM dominate_VVI the_AT spectral_JJ matrix_NN1 ;_; or_CC if_CS they_PPHS2 are_VBR very_RG large_JJ ,_, a_AT1 unit_NN1 could_VM be_VBI of_IO the_AT order_NN1 of_IO the_AT errors_NN2 in_II the_AT computation_NN1 ._. 
It_PPH1 would_VM then_RT be_VBI better_JJR to_TO choose_VVI a_AT1 quantity_NN1 of_IO the_AT order_NN1 of_IO magnitude_NN1 of_IO the_AT repeated_JJ eigenvalue_NN1 ,_, or_CC even_RR the_AT eigenvalue_NN1 itself_PPX1 ,_, provided_CS this_DD1 is_VBZ not_XX zero._NNU 1.21_MC SOME_DD REMARKS_NN2 ON_II EQUAL_JJ EIGENVALUES_NN2 We_PPIS2 have_VH0 already_RR remarked_VVN that_CST the_AT case_NN1 of_IO equal_JJ or_CC confluent_JJ eigenvalues_NN2 is_VBZ not_XX common_JJ in_II practical_JJ dynamical_JJ problems_NN2 ;_; however_RR ,_, when_CS it_PPH1 occurs_VVZ ,_, it_PPH1 can_VM cause_VVI difficulties_NN2 ._. 
We_PPIS2 shall_VM not_XX attempt_VVI to_TO deal_VVI with_IW the_AT full_JJ problem_NN1 of_IO multiple_JJ equal_JJ eigenvalues_NN2 ;_; it_PPH1 will_VM be_VBI sufficient_JJ in_RR21 general_RR22 to_TO consider_VVI the_AT case_NN1 of_IO two_MC equal_JJ roots_NN2 &lsqb;_( formula_NN1 &rsqb;_) 1_MC1 =_FO &lsqb;_( formula_NN1 &rsqb;_) 2_MC as_CSA typical_JJ ._. 
Extension_NN1 to_II more_RGR complicated_JJ cases_NN2 is_VBZ not_XX difficult_JJ (_( a_ZZ1 )_) Extension_NN1 of_IO the_AT Cayley-Hamilton_NP1 theorem_NN1 In_II 1.17_MC we_PPIS2 showed_VVD that_CST a_AT1 matrix_NN1 A_ZZ1 satisfies_VVZ its_APPGE own_DA characteristic_JJ equation_NN1 ,_, i.e._REX &lsqb;_( formula_NN1 &rsqb;_) irrespective_II21 of_II22 the_AT nature_NN1 of_IO the_AT eigenvalues_NN2 ._. 
Now_RT when_RRQ &lsqb;_( formula_NN1 &rsqb;_) ,_, the_AT characteristic_JJ matrix_NN1 (_( &lsqb;_( formula_NN1 &rsqb;_) I_ZZ1 A_ZZ1 )_) may_VM be_VBI simply_RR or_CC doubly_RR degenerate_JJ ,_, according_II21 to_II22 the_AT nature_NN1 of_IO its_APPGE canonical_JJ form_NN1 &lsqb;_( formula_NN1 &rsqb;_) If_CS it_PPH1 is_VBZ simply_RR degenerate_JJ (_( i.e._REX its_APPGE canonical_JJ form_NN1 includes_VVZ a_AT1 unit_NN1 in_II the_AT superdiagonal_JJ )_) then_RT &lsqb;_( formula_NN1 &rsqb;_) (_( A_ZZ1 )_) =_FO 0_MC is_VBZ in_II fact_NN1 the_AT vanishing_JJ polynomial_NN1 in_II A_ZZ1 of_IO lowest_JJT degree_NN1 ._. 
But_CCB if_CS it_PPH1 is_VBZ doubly_RR degenerate_JJ ,_, there_EX is_VBZ a_AT1 reduced_JJ characteristic_JJ equation_NN1 &lsqb;_( formula_NN1 &rsqb;_) being_VBG one_MC1 degree_NN1 lower_JJR than_CSN &lsqb;_( formula_NN1 &rsqb;_) ._. 
Consider_VV0 the_AT equation_NN1 &lsqb;_( formula_NN1 &rsqb;_) where_RRQ &lsqb;_( formula_NN1 &rsqb;_) is_VBZ the_AT adjugate_NN1 of_IO (_( &lsqb;_( formula_NN1 &rsqb;_) if_CS for_IF &lsqb;_( formula_NN1 &rsqb;_) (_( &lsqb;_( formula_NN1 &rsqb;_) is_VBZ doubly_RR degenerate_JJ &lsqb;_( formula_NN1 &rsqb;_) is_VBZ null_JJ ,_, since_CS all_DB first_MD minors_NN2 of_IO (_( &lsqb;_( formula_NN1 &rsqb;_) I_ZZ1 A_ZZ1 )_) vanish_VV0 ._. 
Accordingly_RR ,_, &lsqb;_( formula_NN1 &rsqb;_) must_VM have_VHI the_AT factor_NN1 (_( &lsqb;_( formula_NN1 &rsqb;_) &lsqb;_( formula_NN1 &rsqb;_) in_II each_DD1 element_NN1 ,_, while_CS &lsqb;_( formula_NN1 &rsqb;_) (_( &lsqb;_( formula_NN1 &rsqb;_) )_) has_VHZ two_MC such_DA factors_NN2 ._. 
If_CS we_PPIS2 cancel_VV0 one_MC1 factor_NN1 from_II both_DB2 sides_NN2 ,_, we_PPIS2 can_VM write_VVI &lsqb;_( formula_NN1 &rsqb;_) and_CC then_RT ,_, as_CSA before_RT ,_, we_PPIS2 can_VM prove_VVI that_CST &lsqb;_( formula_NN1 &rsqb;_) ._. 
An_AT1 alternative_JJ expression_NN1 is_VBZ evidently_RR &lsqb;_( formula_NN1 &rsqb;_) where_RRQ the_AT first_MD factor_NN1 is_VBZ doubly_RR degenerate_JJ ;_; the_AT remainder_NN1 are_VBR simply_RR degenerate_JJ ,_, giving_VVG as_CSA before_II a_AT1 total_JJ degeneracy_NN1 n_ZZ1 ._. 
But_CCB if_CS the_AT first_MD factor_NN1 is_VBZ simply_RR degenerate_JJ ,_, it_PPH1 must_VM appear_VVI twice_RR ._. 
We_PPIS2 may_VM illustrate_VVI this_DD1 with_IW a_AT1 simple_JJ numerical_JJ case_NN1 ._. 
Let_VV0 &lsqb;_( formula_NN1 &rsqb;_) Put_VV0 &lsqb;_( formula_NN1 &rsqb;_) =_FO 2_MC and_CC &lsqb;_( formula_NN1 &rsqb;_) in_II the_AT characteristic_JJ matrix_NN1 &lsqb;_( formula_NN1 &rsqb;_) then_RT &lsqb;_( formula_NN1 &rsqb;_) since_CS by_II inspection_NN1 it_PPH1 is_VBZ doubly_RR degenerate_JJ ._. 
Also_RR &lsqb;_( formula_NN1 &rsqb;_) which_DDQ is_VBZ simply_RR degenerate_JJ ;_; the_AT sum_NN1 of_IO the_AT columns_NN2 vanishes_VVZ ,_, but_CCB first_MD minors_NN2 do_VD0 not_XX ._. 
By_II inspection_NN1 &lsqb;_( formula_NN1 &rsqb;_) satisfying_VVG the_AT reduced_JJ characteristic_JJ equation_NN1 ._. 
However_RR ,_, when_CS we_PPIS2 put_VV0 &lsqb;_( formula_NN1 &rsqb;_) &lsqb;_( formula_NN1 &rsqb;_) in_II the_AT characteristic_JJ matrix_NN1 &lsqb;_( formula_NN1 &rsqb;_) ,_, we_PPIS2 obtain_VV0 &lsqb;_( formula_NN1 &rsqb;_) both_DB2 of_IO which_DDQ are_VBR simply_RR degenerate_JJ ;_; it_PPH1 is_VBZ readily_RR checked_VVN that_CST &lsqb;_( formula_NN1 &rsqb;_) which_DDQ satisfies_VVZ the_AT full_JJ characteristic_JJ equation_NN1 ._. 
If_CS a_AT1 reduced_JJ characteristic_JJ equation_NN1 exists_VVZ ,_, so_CS21 that_CS22 &lsqb;_( formula_NN1 &rsqb;_) ,_, then_RT &lsqb;_( formula_NN1 &rsqb;_) ..._... also_RR vanish_VV0 ;_; and_CC ,_, as_CSA in_II 1.1_MC ,_, we_PPIS2 may_VM now_RT express_VVI powers_NN2 of_IO A_ZZ1 or_CC polynomials_NN2 in_II A_ZZ1 as_CSA polynomials_NN2 in_II A_ZZ1 of_IO degree_NN1 one_MC1 less_DAR than_CSN that_DD1 of_IO &lsqb;_( formula_NN1 &rsqb;_) ._. 
(_( b_ZZ1 )_) Nature_NN1 of_IO modal_JJ matrix_NN1 We_PPIS2 now_RT turn_VV0 to_II the_AT confluent_JJ form_NN1 of_IO Equation_NN1 (_( 1.16.7_MC )_) ,_, viz._REX AX_NP1 =_FO XA_NP1 ,_, and_CC we_PPIS2 shall_VM deal_VVI with_IW this_DD1 by_II using_VVG the_AT example_NN1 of_IO the_AT numerical_JJ matrices_NN2 A_ZZ1 and_CC B_ZZ1 above_RL ._. 
By_II inspection_NN1 of_IO (_( 3_MC )_) we_PPIS2 can_VM see_VVI that_CST a_AT1 column_NN1 vector_NN1 xi_NN1 postmultiplying_VVG (_( 2I_FO A_ZZ1 )_) to_TO give_VVI a_AT1 null_JJ result_NN1 must_VM be_VBI such_CS21 that_CS22 &lsqb;_( formula_NN1 &rsqb;_) vanishes_VVZ ;_; two_MC obvious_JJ vectors_NN2 achieving_VVG this_DD1 are_VBR 1,2,4_MC )_) and_CC 1,0_MC ,_, 1_MC1 ._. 
Also_RR if_CS (_( I_ZZ1 A_ZZ1 )_) is_VBZ postmultiplied_VVN by_II 1,1,1_MC it_PPH1 vanishes_VVZ ._. 
Hence_RR we_PPIS2 may_VM write_VVI (_( 8_MC )_) as_CSA &lsqb;_( formula_NN1 &rsqb;_) so_CS21 that_CS22 ,_, in_II this_DD1 case_NN1 (_( two_MC equal_JJ roots_NN2 and_CC double_JJ degeneracy_NN1 of_IO the_AT characteristic_JJ matrix_NN1 )_) ,_, the_AT form_NN1 of_IO (_( 8_MC )_) is_VBZ unaltered_JJ ._. 
However_RR ,_, it_PPH1 should_VM be_VBI noted_VVN that_CST ,_, while_CS the_AT first_MD two_MC columns_NN2 of_IO X_ZZ1 are_VBR arbitrary_JJ to_II a_AT1 scalar_JJ multiplier_NN1 ,_, they_PPHS2 are_VBR also_RR obviously_RR arbitrary_JJ to_II any_DD linear_JJ combination_NN1 ;_; i.e._REX we_PPIS2 may_VM postmultiply_RR the_AT equation_NN1 by_II a_AT1 matrix_NN1 &lsqb;_( formula_NN1 &rsqb;_) in_II which_DDQ &lsqb;_( formula_NN1 &rsqb;_) and_CC &lsqb;_( formula_NN1 &rsqb;_) are_VBR arbitrary_JJ except_CS21 that_CS22 D_ZZ1 must_VM be_VBI non-singular_JJ ._. 
The_AT leading_JJ second-order_JJ submatrix_NN1 of_IO D_ZZ1 corresponds_VVZ to_II the_AT scalar_JJ submatrix_NN1 in_II A_ZZ1 and_CC therefore_RR permutes_VVZ with_IW it_PPH1 ,_, so_CS21 that_CS22 as_CSA before_II &lsqb;_( formula_NN1 &rsqb;_) and_CC XD_NP1 is_VBZ a_AT1 solution_NN1 of_IO (_( 8_MC )_) ._. 
Now_RT consider_VV0 the_AT matrix_NN1 B_ZZ1 ,_, which_DDQ yields_VVZ two_MC equal_JJ eigenvalues_NN2 but_CCB simple_JJ degeneracy_NN1 ._. 
Since_CS 2I_FO B_ZZ1 and_CC I_ZZ1 B_ZZ1 are_VBR both_RR simply_RR degenerate_JJ we_PPIS2 can_VM find_VVI only_RR one_MC1 vector_NN1 for_IF each_DD1 to_TO satisfy_VVI &lsqb;_( formula_NN1 &rsqb;_) ;_; they_PPHS2 are_VBR respectively_RR (_( see_VV0 (_( 6_MC )_) )_) 1,2,4_MC and_CC 1.1.1_MC ._. 
No_AT other_JJ vector_NN1 (_( apart_II21 from_II22 scalar_JJ multiplies_VVZ )_) satisfies_VVZ the_AT equation_NN1 ._. 
It_PPH1 follows_VVZ that_CST we_PPIS2 can_VM not_XX make_VVI up_RP a_AT1 square_JJ matrix_NN1 X_ZZ1 of_IO eigenvectors_NN2 to_TO satisfy_VVI (_( 8_MC )_) ;_; for_IF this_DD1 reason_NN1 the_AT matrix_NN1 B_ZZ1 is_VBZ said_VVN to_TO be_VBI defective_JJ ._. 
However_RR ,_, we_PPIS2 may_VM note_VVI that_CST 1.2.4_MC and_CC 1.1.1_MC are_VBR the_AT first_MD and_CC third_MD columns_NN2 of_IO the_AT modal_JJ matrix_NN1 X_ZZ1 appropriate_JJ to_II A._NNU If_CS we_PPIS2 try_VV0 this_DD1 X_ZZ1 for_IF B_ZZ1 we_PPIS2 find_VV0 that_CST we_PPIS2 obtain_VV0 L_ZZ1 or_CC &lsqb;_( formula_NN1 &rsqb;_) where_RRQ &lsqb;_( formula_NN1 &rsqb;_) is_VBZ now_RT the_AT diagonal_JJ matrix_NN1 of_IO the_AT eigenvalues_NN2 with_IW the_AT addition_NN1 of_IO a_AT1 unit_NN1 in_II the_AT top_JJ superdiagonal_JJ place_NN1 ._. 
Evidently_RR A_ZZ1 is_VBZ the_AT canonical_JJ form_NN1 of_IO A_ZZ1 ,_, and_CC &lsqb;_( formula_NN1 &rsqb;_) that_DD1 of_IO the_AT defective_JJ matrix_NN1 B._NP1 In_II this_DD1 latter_DA case_NN1 ,_, a_AT1 postmultiplying_JJ non-singular_JJ matrix_NN1 analogous_JJ to(9)_FO which_DDQ will_VM permute_VVI with_IW &lsqb;_( formula_NN1 &rsqb;_) is_VBZ &lsqb;_( formula_NN1 &rsqb;_) where_RRQ &lsqb;_( formula_NN1 &rsqb;_) and_CC &lsqb;_( formula_NN1 &rsqb;_) are_VBR otherwise_RR arbitrary_JJ ._. 
Then_RT &lsqb;_( formula_NN1 &rsqb;_) satisfies_VVZ (_( 10_MC )_) ._. 
We_PPIS2 may_VM note_VVI that_DD1 ,_, for_IF both_DB2 the_AT matrices_NN2 A_ZZ1 and_CC B_ZZ1 ,_, &lsqb;_( formula_NN1 &rsqb;_) we_PPIS2 may_VM therefore_RR construct_VVI the_AT canonical_JJ forms_NN2 directly_RR from_II &lsqb;_( formula_NN1 &rsqb;_) We_PPIS2 now_RT revert_VV0 to_II (_( 10_MC )_) ;_; if_CS for_IF convenience_NN1 we_PPIS2 denote_VV0 the_AT columns_NN2 of_IO X_ZZ1 by_II &lsqb;_( formula_NN1 &rsqb;_) respectively_RR ,_, then_RT as_CSA we_PPIS2 have_VH0 said_VVN above_RL ,_, only_RR x1_FO and_CC x3_FO are_VBR eigenvectors_NN2 ;_; they_PPHS2 make_VV0 &lsqb;_( formula_NN1 &rsqb;_) and_CC &lsqb;_( formula_NN1 &rsqb;_) vanish_VV0 ,_, and_CC are_VBR each_DD1 unique_JJ to_II a_AT1 scalar_JJ multiplier_NN1 ._. 
We_PPIS2 now_RT examine_VV0 the_AT origins_NN2 of_IO &lsqb;_( formula_NN1 &rsqb;_) ,_, which_DDQ in_II31 relation_II32 to_II33 B_ZZ1 we_PPIS2 call_VV0 an_AT1 auxiliary_JJ vector_NN1 ._. 
In_II (_( 10_MC )_) we_PPIS2 find_VV0 &lsqb;_( formula_NN1 &rsqb;_) Premultiplication_NN1 of_IO this_DD1 by_II (_( 2I_FO B_ZZ1 )_) yields_NN2 &lsqb;_( formula_NN1 &rsqb;_) Accordingly_RR ,_, postmultiplication_NN1 by_II the_AT auxiliary_JJ vector_NN1 &lsqb;_( formula_NN1 &rsqb;_) annihilates_NN2 &lsqb;_( formula_NN1 &rsqb;_) ,_, in_II fact_NN1 ,_, since_CS &lsqb;_( formula_NN1 &rsqb;_) is_VBZ doubly_RR degenerate_JJ ,_, there_EX will_VM be_VBI two_MC linearly_RR independent_JJ postmultiplying_JJ vectors_NN2 which_DDQ annihilate_VV0 it_PPH1 ,_, separately_RR or_CC in_II linear_JJ combinations_NN2 ._. 
We_PPIS2 can_VM ,_, and_CC indeed_RR must_VM ,_, choose_VV0 x1_FO as_CSA one_MC1 of_IO these_DD2 ,_, for_IF sine_FW &lsqb;_( formula_NN1 &rsqb;_) vanishes_VVZ ,_, so_RG evidently_RR does_VDZ &lsqb;_( formula_NN1 &rsqb;_) ;_; and_CC then_RT &lsqb;_( formula_NN1 &rsqb;_) is_VBZ the_AT other_JJ ._. 
We_PPIS2 might_VM have_VHI anticipated_VVN this_DD1 result_NN1 from_II Equations_NN2 (_( 5_MC )_) and_CC (_( 7_MC )_) ._. 
I_ZZ1 (_( 5_MC )_) the_AT doubly_RR degenerate_JJ matrix_NN1 (_( 2I_FO -A_ZZ1 )_) is_VBZ annihilated_VVN by_II two_MC linearly_RR independent_JJ eigenvectors_NN2 and_CC the_AT simply_RR degenerate_JJ (_( I_ZZ1 A_ZZ1 )_) by_II the_AT third_MD ._. 
In_II (_( 7_MC )_) ,_, &lsqb;_( formula_NN1 &rsqb;_) is_VBZ doubly_RR degenerate_JJ and_CC so_RR corresponds_VVZ with_IW (_( 2I_FO A_ZZ1 )_) ;_; however_RR ,_, in_II this_DD1 case_NN1 one_MC1 eigenvector_NN1 and_CC one_MC1 auxiliary_JJ vector_NN1 emerge_VV0 ._. 
It_PPH1 is_VBZ now_RT clear_VV0 that_CST ,_, to_TO make_VVI up_RP X_ZZ1 ,_, we_PPIS2 require_VV0 (_( i_ZZ1 )_) an_AT1 eigenvector_NN1 x1_FO satisfying_JJ &lsqb;_( formula_NN1 &rsqb;_) (_( ii_MC )_) an_AT1 auxiliary_JJ vector_NN1 &lsqb;_( formula_NN1 &rsqb;_) satisfying_JJ &lsqb;_( formula_NN1 &rsqb;_) (_( iii_MC )_) an_AT1 eigenvector_NN1 x3_FO satisfying_JJ &lsqb;_( formula_NN1 &rsqb;_) In_II the_AT numerical_JJ example_NN1 (_( 10_MC )_) ,_, &lsqb;_( formula_NN1 &rsqb;_) and_CC postmultiplication_NN1 by_II x1_FO =_FO 1,2,4_MC makes_VVZ both_RR vanish_VV0 ;_; postmultiplication_NN1 by_II &lsqb;_( formula_NN1 &rsqb;_) =_FO 1,0_MC ,_, -1_MC annihilates_VVZ the_AT second_NNT1 but_CCB not_XX the_AT first_MD ;_; the_AT product_NN1 yields_VVZ x1_FO ._. 
There_EX is_VBZ one_MC1 further_JJR point_NN1 of_IO importance_NN1 :_: &lsqb;_( formula_NN1 &rsqb;_) is_VBZ not_XX unique_JJ ,_, but_CCB contains_VVZ an_AT1 arbitrary_JJ element_NN1 ._. 
As_CSA we_PPIS2 have_VH0 seen_VVN ,_, &lsqb;_( formula_NN1 &rsqb;_) is_VBZ satisfied_VVN by_II two_MC linearly_RR independent_JJ vectors_NN2 (_( in_II our_APPGE example_NN1 ,_, 1,2,4_MC and_CC 1,0_MC ,_, -1_MC and_CC therefore_RR by_II any_DD linear_JJ combinations_NN2 of_IO them_PPHO2 ._. 
One_PN1 is_VBZ chosen_VVN to_TO be_VBI x1_FO ,_, satisfying_JJ (_( 2I_FO B_ZZ1 )_) x_ZZ1 =_FO 0_MC ;_; the_AT other_JJ &lsqb;_( formula_NN1 &rsqb;_) ,_, is_VBZ another_DD1 arbitrary_JJ combination_NN1 ._. 
However_RR ,_, it_PPH1 must_VM satisfy_VVI &lsqb;_( formula_NN1 &rsqb;_) so_CS21 that_CS22 if_CS x1_FO is_VBZ multiplied_VVN by_II a_AT1 scalar_JJ d1_FO so_RG also_RR is_VBZ &lsqb;_( formula_NN1 &rsqb;_) ;_; which_DDQ is_VBZ why_RRQ (_( 11_MC )_) contains_VVZ equal_JJ elements_NN2 &lsqb;_( formula_NN1 &rsqb;_) ._. 
But_CCB the_AT equation_NN1 is_VBZ also_RR satisfied_VVN by_II &lsqb;_( formula_NN1 &rsqb;_) so_CS21 that_CS22 in_II our_APPGE example_NN1 &lsqb;_( formula_NN1 &rsqb;_) with_IW &lsqb;_( formula_NN1 &rsqb;_) arbitrary_JJ ._. 
This_DD1 result_NN1 is_VBZ evidently_RR generally_RR true_JJ ._. 
As_II a_AT1 dynamical_JJ example_NN1 ,_, perhaps_RR the_AT commonest_JJT practical_JJ problem_NN1 leading_VVG to_II a_AT1 double_JJ eigenvalue_NN1 with_IW simple_JJ degeneracy_NN1 is_VBZ that_DD1 of_IO critical_JJ damping_NN1 ._. 
For_REX21 example_REX22 ,_, the_AT equation_NN1 of_IO motion_NN1 of_IO a_AT1 mass_NN1 constrained_VVN by_II a_AT1 spring_NN1 and_CC dashpot_NN1 is_VBZ ,_, in_II the_AT well-known_JJ standard_JJ form_NN1 ,_, &lsqb;_( formula_NN1 &rsqb;_) Here_RL we_PPIS2 shall_VM consider_VVI &lsqb;_( formula_NN1 &rsqb;_) as_CSA fixed_JJ and_CC &lsqb;_( formula_NN1 &rsqb;_) as_II a_AT1 variable_NN1 (_( in_II practice_NN1 ,_, a_AT1 variable_JJ spring_NN1 stiffness_NN1 )_) and_CC for_IF convenience_NN1 we_PPIS2 shall_VM write_VVI &lsqb;_( formula_NN1 &rsqb;_) for_IF &lsqb;_( formula_NN1 &rsqb;_) ;_; this_DD1 &lsqb;_( formula_NN1 &rsqb;_) is_VBZ a_AT1 variable_JJ parameter_NN1 ,_, which_DDQ may_VM be_VBI positive_JJ or_CC negative_JJ :_: in_II the_AT latter_DA case_NN1 ,_, the_AT motion_NN1 is_VBZ a_AT1 damped_JJ oscillation_NN1 ._. 
If_CS x_ZZ1 is_VBZ proportional_JJ to_TO exp&lsqb;formula&rsqb;t_VVI ,_, we_PPIS2 have_VH0 to_TO satisfy_VVI &lsqb;_( formula_NN1 &rsqb;_) and_CC for_IF &lsqb;_( formula_NN1 &rsqb;_) we_PPIS2 therefore_RR have_VH0 two_MC different_JJ roots_NN2 :_: either_RR two_MC real_JJ roots_NN2 or_CC a_AT1 conjugate_NN1 complex_JJ pair_NN ._. 
However_RR ,_, for_IF &lsqb;_( formula_NN1 &rsqb;_) (_( the_AT critical_JJ damping_NN1 case_NN1 )_) the_AT two_MC roots_NN2 coalesce_VV0 into_II a_AT1 repeated_JJ (_( real_JJ )_) root_NN1 ,_, &lsqb;_( formula_NN1 &rsqb;_) =_FO &lsqb;_( formula_NN1 &rsqb;_) If_CS we_PPIS2 write_VV0 v_ZZ1 for_IF x_ZZ1 ,_, the_AT equation_NN1 of_IO motion_NN1 may_VM be_VBI written_VVN as_II the_AT first-order_JJ pair_NN &lsqb;_( formula_NN1 &rsqb;_) or_CC ,_, when_CS x_ZZ1 ,_, v_ZZ1 is_VBZ proportional_JJ to_TO exp&lsqb;formula&rsqb;t_VVI ,_, &lsqb;_( formula_NN1 &rsqb;_) This_DD1 requires_VVZ &lsqb;_( formula_NN1 &rsqb;_) Giving_VVG of_RR21 course_RR22 the_AT eigenvalues_NN2 &lsqb;_( formula_NN1 &rsqb;_) for_IF each_DD1 of_IO which_DDQ &lsqb;_( formula_NN1 &rsqb;_) is_VBZ simply_RR degenerate_JJ ._. 
In_II the_AT critical_JJ case_NN1 ,_, when_CS ge_NN1 vanishes_VVZ ,_, we_PPIS2 have_VH0 two_MC equal_JJ eigenvalues_NN2 -gm_NN1 ,_, but_CCB &lsqb;_( formula_NN1 &rsqb;_) I_ZZ1 B(0)_FO is_VBZ still_RR simply_RR degenerate_JJ ;_; thus_RR &lsqb;_( formula_NN1 &rsqb;_) and_CC so_RR the_AT column_NN1 &lsqb;_( formula_NN1 &rsqb;_) makes_VVZ this_DD1 vanish_VV0 ._. 
Squaring_VVG it_PPH1 ,_, we_PPIS2 obtain_VV0 a_AT1 null_JJ matrix_NN1 ,_, so_CS21 that_CS22 &lsqb;_( formula_NN1 &rsqb;_) is_VBZ satisfied_VVN by_II two_MC linearly_RR independent_JJ arbitrary_JJ columns_NN2 ._. 
As_CSA we_PPIS2 have_VH0 seen_VVN ,_, one_MC1 of_IO them_PPHO2 must_VM be_VBI &lsqb;_( formula_NN1 &rsqb;_) the_AT most_RGT general_JJ form_NN1 of_IO the_AT other_NN1 is_VBZ &lsqb;_( formula_NN1 &rsqb;_) ._. 
Choosing_NN1 &lsqb;_( formula_NN1 &rsqb;_) we_PPIS2 find_VV0 Equation_NN1 (_( 13_MC )_) to_TO be_VBI ,_, in_II this_DD1 case_NN1 ,_, &lsqb;_( formula_NN1 &rsqb;_) with_IW the_AT spectral_JJ matrix_NN1 (_( on_II the_AT right_NN1 )_) in_II the_AT standard_JJ canonical_JJ form_NN1 for_IF a_AT1 defective_JJ matrix_NN1 ._. 
There_EX is_VBZ a_AT1 further_JJR interpretation_NN1 of_IO this_DD1 which_DDQ is_VBZ of_IO great_JJ importance_NN1 ._. 
With_IW &lsqb;_( formula_NN1 &rsqb;_) general_NN1 ,_, we_PPIS2 may_VM write_VVI the_AT eigenvectors_NN2 corresponding_VVG to_II &lsqb;_( formula_NN1 &rsqb;_) ,_, &lsqb;_( formula_NN1 &rsqb;_) respectively_RR ;_; hence_RR the_AT operation_NN1 &lsqb;_( formula_NN1 &rsqb;_) and_CC &lsqb;_( formula_NN1 &rsqb;_) is_VBZ converted_VVN to_II the_AT standard_JJ canonical_JJ form_NN1 for_IF matrices_NN2 with_IW distinct_JJ eigenvalues_NN2 ._. 
However_RR ,_, this_DD1 similar_JJ transformation_NN1 fails_VVZ when_RRQ &lsqb;_( formula_NN1 &rsqb;_) :_: the_AT two_MC columns_NN2 of_IO X(O)_NP1 become_VV0 identical_JJ ,_, so_CS21 that_CS22 its_APPGE reciprocal_JJ becomes_VVZ infinite_JJ ._. 
However_RR ,_, if_CS we_PPIS2 postmultiply_RR &lsqb;_( formula_NN1 &rsqb;_) by_II a_AT1 matrix_NN1 which_DDQ in_II effect_NN1 sums_NN2 and_CC differences_NN2 the_AT columns_NN2 ,_, thus_RR :_: &lsqb;_( formula_NN1 &rsqb;_) then_RT a_AT1 similar_JJ transformation_NN1 &lsqb;_( formula_NN1 &rsqb;_) yields_NN2 &lsqb;_( formula_NN1 &rsqb;_) as_CSA follows_VVZ :_: &lsqb;_( formula_NN1 &rsqb;_) i.e._REX a_AT1 quasi-canonical_JJ form_NN1 :_: it_PPH1 is_VBZ to_TO be_VBI remembered_VVN that_CST the_AT standard_JJ canonical_JJ forms_NN2 are_VBR conventional_JJ ,_, but_CCB not_XX unique_JJ ._. 
The_AT two_MC matrices_NN2 &lsqb;_( formula_NN1 &rsqb;_) and_CC &lsqb;_( formula_NN1 &rsqb;_) are_VBR both_RR equivalent_JJ to_II &lsqb;_( formula_NN1 &rsqb;_) and_CC to_II each_PPX221 other_PPX222 ;_; they_PPHS2 are_VBR each_DD1 derived_VVN from_II &lsqb;_( formula_NN1 &rsqb;_) by_II a_AT1 similar_JJ transformation_NN1 ._. 
But_CCB the_AT first_MD is_VBZ possible_JJ only_RR for_IF &lsqb;_( formula_NN1 &rsqb;_) ,_, while_CS the_AT second_MD becomes_VVZ a_AT1 canonical_JJ form_NN1 appropriate_JJ to_II a_AT1 defective_JJ matrix_NN1 only_RR for_IF &lsqb;_( formula_NN1 &rsqb;_) This_DD1 example_NN1 shows_VVZ that_CST the_AT auxiliary_JJ vector_NN1 0,1_MC is_VBZ in_II effect_NN1 the_AT differential_NN1 with_II31 respect_II32 to_II33 &lsqb;_( formula_NN1 &rsqb;_) of_IO the_AT two_MC confluent_JJ vectors_NN2 &lsqb;_( formula_NN1 &rsqb;_) and_CC &lsqb;_( formula_NN1 &rsqb;_) ._. 
(_( c_ZZ1 )_) Extension_NN1 of_IO Sylvester_NP1 's_GE expansion_NN1 We_PPIS2 come_VV0 finally_RR to_II the_AT confluent_JJ form_NN1 of_IO Sylvester_NP1 's_GE expansion_NN1 for_IF any_DD polynomial_NN1 ,_, and_CC again_RT we_PPIS2 use_VV0 the_AT matrices_NN2 A_ZZ1 and_CC B_ZZ1 as_CSA exemplifying_VVG the_AT general_JJ case_NN1 ;_; however_RR ,_, for_IF greater_JJR generality_NN1 we_PPIS2 shall_VM write_VVI the_AT repeated_JJ roots_NN2 as_CSA &lsqb;_( formula_NN1 &rsqb;_) the_AT unrepeated_JJ root_NN1 as_CSA &lsqb;_( formula_NN1 &rsqb;_) and_CC shall_VM replace_VVI the_AT unit_NN1 in_II the_AT superdiagonal_JJ of_IO &lsqb;_( formula_NN1 &rsqb;_) by_II r_ZZ1 ._. 
Them_PPHO2 ,_, treating_VVG A_AT1 first_MD ,_, we_PPIS2 have_VH0 as_RG in_II 1.18_MC &lsqb;_( formula_NN1 &rsqb;_) The_AT expansion_NN1 can_VM then_RT proceed_VVI as_CSA before_RT ,_, and_CC we_PPIS2 obtain_VV0 &lsqb;_( formula_NN1 &rsqb;_) where_RRQ &lsqb;_( formula_NN1 &rsqb;_) are_VBR unit_NN1 rank_NN1 matrices_NN2 ._. 
The_AT case_NN1 of_IO B_ZZ1 ,_, involving_VVG a_AT1 double_JJ root_NN1 with_IW simple_JJ degeneracy_NN1 ,_, is_VBZ a_RR21 little_RR22 more_RGR complicated_JJ ._. 
In_II succession_NN1 ,_, we_PPIS2 find_VV0 &lsqb;_( formula_NN1 &rsqb;_) Now_RT P_ZZ1 (_( &lsqb;_( formula_NN1 &rsqb;_) is_VBZ the_AT polynomial_NN1 in_II &lsqb;_( formula_NN1 &rsqb;_) &lsqb;_( formula_NN1 &rsqb;_) which_DDQ on_II differentiation_NN1 gives_VVZ &lsqb;_( formula_NN1 &rsqb;_) We_PPIS2 can_VM evidently_RR construct_VVI corresponding_JJ polynomials_NN2 in_II the_AT matrices_NN2 B_ZZ1 and_CC &lsqb;_( formula_NN1 &rsqb;_) ,_, and_CC in_II31 view_II32 of_II33 (_( 13_MC )_) they_PPHS2 will_VM be_VBI connected_VVN by_II &lsqb;_( formula_NN1 &rsqb;_) in_II (_( 17_MC )_) ,_, the_AT first_MD column_NN1 of_IO X_ZZ1 is_VBZ proportion_NN1 to_II &lsqb;_( formula_NN1 &rsqb;_) ._. 
Using_VVG (_( 15_MC )_) ,_, we_PPIS2 may_VM expand_VVI &lsqb;_( formula_NN1 &rsqb;_) in_II the_AT form_NN1 &lsqb;_( formula_NN1 &rsqb;_) Accordingly_RR ,_, if_CS we_PPIS2 now_RT proceed_VV0 (_( exactly_RR as_CSA in_II 1.18_MC )_) to_II premultiply_RR (_( 17_MC )_) by_II X_ZZ1 and_CC postmultiply_RR by_II &lsqb;_( formula_NN1 &rsqb;_) (_( &lsqb;_( formula_NN1 &rsqb;_) ;_; the_AT first_MD row_NN1 of_IO &lsqb;_( formula_NN1 &rsqb;_) is_VBZ proportional_JJ to_II r_ZZ1 )_) we_PPIS2 obtain_VV0 &lsqb;_( formula_NN1 &rsqb;_) Comparison_NN1 with_IW (_( 14_MC )_) shows_VVZ that_CST we_PPIS2 have_VH0 now_RT added_VVN a_AT1 new_JJ term_NN1 &lsqb;_( formula_NN1 &rsqb;_) ,_, which_DDQ derives_VVZ from_II the_AT element_NN1 r_ZZ1 in_II the_AT superdiagonal_JJ of_IO the_AT canonical_JJ form_NN1 ._. 
This_DD1 also_RR is_VBZ a_AT1 square_JJ matrix_NN1 of_IO unit_NN1 rank_NN1 ,_, and_CC z12_FO evidently_RR has_VHZ the_AT properties_NN2 &lsqb;_( formula_NN1 &rsqb;_) In_II fact_NN1 ,_, the_AT above_JJ equations_NN2 ,_, and_CC the_AT properties_NN2 of_IO the_AT matrices_NN2 &lsqb;_( formula_NN1 &rsqb;_) discussed_VVD in_II 1.18_MC ,_, are_VBR all_DB contained_VVN in_II the_AT general_JJ result_NN1 &lsqb;_( formula_NN1 &rsqb;_) on_II evaluation_NN1 of_IO the_AT inner_JJ products_NN2 ._. 
Equation_NN1 (_( 18_MC )_) is_VBZ the_AT confluent_JJ form_NN1 of_IO Sylvester_NP1 's_GE expansion_NN1 for_IF the_AT case_NN1 of_IO two_MC equal_JJ eigenvalues_NN2 which_DDQ give_VV0 only_RR simple_JJ degeneracy_NN1 to_II the_AT characteristic_JJ matrix_NN1 ._. 
As_II an_AT1 example_NN1 of_IO its_APPGE consistency_NN1 ,_, if_CS we_PPIS2 put_VV0 P(B)_FO =_FO I_ZZ1 ,_, giving_VVG &lsqb;_( formula_NN1 &rsqb;_) then_RT as_CSA before_II &lsqb;_( formula_NN1 &rsqb;_) while_CS if_CS P(B)_FO =_FO B_ZZ1 ,_, then_RT &lsqb;_( formula_NN1 &rsqb;_) If_CS we_PPIS2 multiply_VV0 (_( 21_MC )_) and_CC (_( 22_MC )_) the_AT result_NN1 repeats_NN2 (_( 22_MC )_) in_II31 view_II32 of_II33 the_AT properties_NN2 of_IO zii_NN2 and_CC (_( 19_MC )_) above_RL ;_; squaring_VVG (_( 22_MC )_) yields_NN2 &lsqb;_( formula_NN1 &rsqb;_) in_II31 accordance_II32 with_II33 (_( 18_MC )_) ._. 
As_II an_AT1 indication_NN1 of_IO Sylvester_NP1 's_GE expansion_NN1 for_IF more_RGR complicated_JJ cases_NN2 ,_, suppose_VV0 we_PPIS2 have_VH0 &lsqb;_( formula_NN1 &rsqb;_) Then_RT if_CS we_PPIS2 evaluate_VV0 &lsqb;_( formula_NN1 &rsqb;_) ,_, it_PPH1 is_VBZ readily_RR found_VVN that_CST the_AT expansion_NN1 in_II this_DD1 case_NN1 will_VM be_VBI &lsqb;_( formula_NN1 &rsqb;_) Further_JJR extensions_NN2 follow_VV0 this_DD1 pattern_NN1 ._. 
Example_NN1 1_MC1 We_PPIS2 use_VV0 B_ZZ1 as_CSA defined_VVN in_II (_( 2_MC )_) together_RL with_IW &lsqb;_( formula_NN1 &rsqb;_) where_RRQ ,_, in_BCL21 order_BCL22 to_TO obtain_VVI r_ZZ1 =_FO 2_MC in_II &lsqb;_( formula_NN1 &rsqb;_) ,_, the_AT first_MD column_NN1 of_IO X_ZZ1 in_II (_( 10_MC )_) has_VHZ been_VBN halved_VVN ,_, and_CC the_AT first_MD row_NN1 of_IO YT_NN1 in_II (_( 12_MC )_) has_VHZ been_VBN doubled_VVN ._. 
These_DD2 matrices_NN2 satisfy_VV0 &lsqb;_( formula_NN1 &rsqb;_) ._. 
Suppose_VV0 we_PPIS2 wish_VV0 to_TO calculate_VVI B-1_FO ._. 
Then_RT in_II (_( 18_MC )_) we_PPIS2 put_VV0 &lsqb;_( formula_NN1 &rsqb;_) with_IW &lsqb;_( formula_NN1 &rsqb;_) Then_RT &lsqb;_( formula_NN1 &rsqb;_) The_AT reader_NN1 is_VBZ invited_VVN to_TO check_VVI that_CST this_DD1 is_VBZ correct._NNU 1.22_MC SOME_DD MISCELLANEOUS_JJ THEOREMS_NN2 We_PPIS2 append_VV0 here_RL some_DD useful_JJ theorems_NN2 and_CC definitions_NN2 :_: Theorem_NN1 1_MC1 The_AT reciprocal_JJ of_IO the_AT transpose_VV0 of_IO a_AT1 matrix_NN1 A_ZZ1 is_VBZ equal_JJ to_II the_AT transpose_VV0 of_IO the_AT reciprocal_JJ of_IO A._NNU For_IF &lsqb;_( formula_NN1 &rsqb;_) ;_; transpose_VV0 to_TO get_VVI &lsqb;_( formula_NN1 &rsqb;_) ._. 
Postmultiply_RR by_II the_AT reciprocal_JJ of_IO AT_II ,_, and_CC &lsqb;_( formula_NN1 &rsqb;_) ._. 
We_PPIS2 shall_VM write_VVI &lsqb;_( formula_NN1 &rsqb;_) ._. 
Corollary_NN1 If_CS A_ZZ1 is_VBZ symmetric_JJ ,_, so_RG also_RR is_VBZ A-1_MC1 ._. 
Example_NN1 1_MC1 If_CS &lsqb;_( formula_NN1 &rsqb;_) Also_RR &lsqb;_( formula_NN1 &rsqb;_) Theorem_NN1 II_MC The_AT trace_NN1 of_IO a_AT1 matrix_NN1 ,_, which_DDQ is_VBZ the_AT sum_NN1 of_IO the_AT elements_NN2 in_II the_AT principal_JJ diagonal_JJ ,_, equals_VVZ the_AT sum_NN1 of_IO the_AT eigenvalues_NN2 ._. 
If_CS the_AT determinant_NN1 &lsqb;_( formula_NN1 &rsqb;_) of_IO order_NN1 n_ZZ1 is_VBZ expanded_VVN in_II the_AT usual_JJ way_NN1 ,_, the_AT first_MD product_NN1 is_VBZ &lsqb;_( formula_NN1 &rsqb;_) times_II its_APPGE cofactor_NN1 ,_, which_DDQ is_VBZ similar_JJ to_II &lsqb;_( formula_NN1 &rsqb;_) but_CCB of_IO order_NN1 1_MC1 ._. 
All_DB other_JJ products_NN2 are_VBR evidently_RR of_IO degree_NN1 n_ZZ1 2_MC in_II &lsqb;_( formula_NN1 &rsqb;_) at_RR21 most_RR22 ._. 
By_II progressive_JJ expansion_NN1 of_IO the_AT diagonal_JJ minors_NN2 it_PPH1 therefore_RR appears_VVZ that_CST the_AT two_MC leading_JJ terms_NN2 in_II the_AT full_JJ expansion_NN1 are_VBR &lsqb;_( formula_NN1 &rsqb;_) Now_RT if_CS we_PPIS2 write_VV0 the_AT determinant_NN1 as_CSA &lsqb;_( formula_NN1 &rsqb;_) the_AT leading_JJ terms_NN2 are_VBR &lsqb;_( formula_NN1 &rsqb;_) Hence_RR &lsqb;_( formula_NN1 &rsqb;_) Example_NN1 II_MC Let_VV0 &lsqb;_( formula_NN1 &rsqb;_) and_CC on_II expansion_NN1 &lsqb;_( formula_NN1 &rsqb;_) Then_RT &lsqb;_( formula_NN1 &rsqb;_) Theorem_NN1 III_MC The_AT product_NN1 of_IO the_AT eigenvalues_NN2 of_IO A_ZZ1 is_VBZ &lsqb;_( formula_NN1 &rsqb;_) ._. 
In_II &lsqb;_( formula_NN1 &rsqb;_) put_VV0 &lsqb;_( formula_NN1 &rsqb;_) ._. 
Then_RT &lsqb;_( formula_NN1 &rsqb;_) ._. 
Hence_RR if_CS an_AT1 eigenvalue_NN1 of_IO A_ZZ1 is_VBZ zero_MC ,_, A_ZZ1 is_VBZ singular_JJ ;_; and_CC conversely_RR if_CS &lsqb;_( formula_NN1 &rsqb;_) ,_, then_RT one_MC1 or_CC more_DAR eigenvalues_NN2 are_VBR zero_MC ._. 
Example_NN1 III_MC If_CS A_ZZ1 is_VBZ the_AT matrix_NN1 in_II Example_NN1 II_MC ,_, then_RT &lsqb;_( formula_NN1 &rsqb;_) Theorem_NN1 IV_MC If_CS a_AT1 scalar_JJ &lsqb;_( formula_NN1 &rsqb;_) is_VBZ added_VVN to_II each_DD1 principal_JJ diagonal_JJ element_NN1 of_IO A_ZZ1 ,_, the_AT eigenvalues_NN2 of_IO A_ZZ1 are_VBR each_DD1 increased_VVN by_II &lsqb;_( formula_NN1 &rsqb;_) ._. 
This_DD1 is_VBZ immediately_RR obvious_JJ from_II the_AT identity_NN1 &lsqb;_( formula_NN1 &rsqb;_) The_AT sum_NN1 of_IO the_AT eigenvalues_NN2 ,_, and_CC tr_JJ A_ZZ1 ,_, are_VBR each_DD1 increased_VVN by_II &lsqb;_( formula_NN1 &rsqb;_) Example_NN1 IV_MC Let_VV0 us_PPIO2 add_VVI -3_MC to_II each_DD1 diagonal_JJ element_NN1 of_IO the_AT matrix_NN1 A_ZZ1 in_II Examples_NN2 II_MC and_CC III_MC ;_; we_PPIS2 obtain_VV0 &lsqb;_( formula_NN1 &rsqb;_) and_CC &lsqb;_( formula_NN1 &rsqb;_) ._. 
Hence_RR ,_, compared_VVN with_IW Example_NN1 I_ZZ1 ,_, the_AT eigenvalues_NN2 have_VH0 each_DD1 been_VBN reduced_VVN by_II &lsqb;_( formula_NN1 &rsqb;_) and_CC &lsqb;_( formula_NN1 &rsqb;_) ._. 
Theorem_NN1 V_II Any_DD matrix_NN1 can_VM be_VBI expressed_VVN as_II the_AT product_NN1 of_IO two_MC symmetric_JJ matrices_NN2 ._. 
The_AT case_NN1 in_II which_DDQ A_ZZ1 is_VBZ non-defective_JJ has_VHZ already_RR been_VBN treated_VVN in_II 1.19_MC ,_, so_CS21 that_CS22 here_RL we_PPIS2 must_VM consider_VVI the_AT defective_JJ case_NN1 ._. 
We_PPIS2 write_VV0 &lsqb;_( formula_NN1 &rsqb;_) Suppose_VV0 we_PPIS2 have_VH0 a_AT1 Jordan_NP1 block_NN1 or_CC blocks_NN2 such_II21 as_II22 &lsqb;_( formula_NN1 &rsqb;_) in_II &lsqb;_( formula_NN1 &rsqb;_) ._. 
If_CS this_DD1 is_VBZ premultiplied_VVN and_CC postmultiplied_VVN by_II the_AT reversing_NN1 matrix_NN1 Js_NP2 ,_, then_RT &lsqb;_( formula_NN1 &rsqb;_) ._. 
Now_RT let_VV0 K_ZZ1 be_VBI the_AT unit_NN1 matrix_NN1 ,_, except_CS21 that_CS22 wherever_RRQV a_AT1 block_NN1 &lsqb;_( formula_NN1 &rsqb;_) occurs_VVZ in_II &lsqb;_( formula_NN1 &rsqb;_) ,_, the_AT corresponding_JJ Is_VBZ is_VBZ replaced_VVN by_II Js_NP2 in_II K._NP1 Then_RT &lsqb;_( formula_NN1 &rsqb;_) We_PPIS2 now_RT revert_VV0 to_II (_( 1_MC1 )_) ,_, postmultiply_RR by_II &lsqb;_( formula_NN1 &rsqb;_) and_CC use_NN1 (_( 2_MC )_) :_: &lsqb;_( formula_NN1 &rsqb;_) on_II use_NN1 of_IO the_AT transpose_VV0 of_IO (_( 1_MC1 )_) ._. 
Now_RT (_( 3_MC )_) shows_VVZ that_CST AXKXT_NP1 equals_VVZ its_APPGE own_DA transpose_VV0 and_CC so_RR is_VBZ symmetric_JJ ,_, says_VVZ S1_FO ._. 
Also_RR XKXT_NP1 is_VBZ symmetric_JJ ,_, says_VVZ S2_FO ._. 
Thus_RR &lsqb;_( formula_NN1 &rsqb;_) is_VBZ expressible_JJ as_II the_AT product_NN1 of_IO two_MC symmetric_JJ matrices_NN2 ._. 
If_CS we_PPIS2 put_VV0 &lsqb;_( formula_NN1 &rsqb;_) and_CC hence_RR K_ZZ1 =_FO I_ZZ1 ,_, the_AT two_MC factors_NN2 in_II (_( 4_MC )_) reduce_VV0 to_II those_DD2 given_VVN in_II 1.19_MC ._. 
This_DD1 establishes_VVZ that_CST such_DA factorisation_NN1 is_VBZ always_RR possible_JJ ._. 
A_AT1 general_JJ numerical_JJ procedure_NN1 ,_, not_XX involving_VVG X_ZZ1 ,_, &lsqb;_( formula_NN1 &rsqb;_) ,_, is_VBZ given_VVN later_RRR in_II 2.10.3_MC ._. 
Example_NN1 V_ZZ1 (_( a_ZZ1 )_) Non-defective_JJ matrix_NN1 let_VV0 A_ZZ1 be_VBI as_CSA defined_VVN in_II (_( 1.21.1_MC )_) ._. 
Then_RT &lsqb;_( formula_NN1 &rsqb;_) assumes_VVZ the_AT numerical_JJ values_NN2 &lsqb;_( formula_NN1 &rsqb;_) In_II this_DD1 case_NN1 &lsqb;_( formula_NN1 &rsqb;_) and_CC &lsqb;_( formula_NN1 &rsqb;_) The_AT reader_NN1 may_VM check_VVI that_CST &lsqb;_( formula_NN1 &rsqb;_) ._. 
(_( b_ZZ1 )_) Defective_JJ matrix_NN1 we_PPIS2 use_VV0 B_ZZ1 as_CSA defined_VVN in_II (_( 1.21.2_MC )_) ._. 
Then_RT &lsqb;_( formula_NN1 &rsqb;_) takes_VVZ the_AT numerical_JJ values_NN2 &lsqb;_( formula_NN1 &rsqb;_) X_MC being_VBG the_AT same_DA as_CSA in_CS21 case_CS22 (_( a_ZZ1 )_) ._. 
Also_RR if_CS &lsqb;_( formula_NN1 &rsqb;_) then_RT &lsqb;_( formula_NN1 &rsqb;_) while_CS &lsqb;_( formula_NN1 &rsqb;_) and_CC again_RT the_AT reader_NN1 may_VM check_VVI that_CST the_AT product_NN1 of_IO the_AT symmetric_JJ matrices_NN2 &lsqb;_( formula_NN1 &rsqb;_) ._. 
Theorem_NN1 VI_MC AT_II can_VM be_VBI obtained_VVN from_II A_ZZ1 by_II a_AT1 similar_JJ transformation_NN1 ._. 
This_DD1 has_VHZ been_VBN proved_VVN incidentally_RR above_RL ;_; (_( 3_MC )_) may_VM be_VBI written_VVN &lsqb;_( formula_NN1 &rsqb;_) Example_NN1 VI_MC We_PPIS2 use_VV0 the_AT matrix_NN1 B_ZZ1 of_IO Example_NN1 V(b)_NP1 above_RL ._. 
Then_RT &lsqb;_( formula_NN1 &rsqb;_) Theorem_NN1 VII_MC If_CS A_ZZ1 is_VBZ any_DD matrix_NN1 of_IO order_NN1 (_( n_ZZ1 p_ZZ1 )_) and_CC B_NP1 any_DD matrix_NN1 of_IO order_NN1 (_( p_ZZ1 n_ZZ1 )_) where_RRQ &lsqb;_( formula_NN1 &rsqb;_) ,_, then_RT the_AT eigenvalues_NN2 of_IO AB_FO are_VBR those_DD2 of_IO BA_NN1 plus_II n_ZZ1 p_ZZ1 zeros_NN2 ._. 
Consider_VV0 the_AT matrix_NN1 product_NN1 &lsqb;_( formula_NN1 &rsqb;_) where_RRQ the_AT matrices_NN2 are_VBR square_JJ and_CC of_IO order_NN1 N_ZZ1 =_FO n_ZZ1 +_FO p_ZZ1 ,_, appropriately_RR partitioned_VVN ._. 
If_CS we_PPIS2 take_VV0 determinants_NN2 and_CC note_VV0 that_CST &lsqb;_( formula_NN1 &rsqb;_) we_PPIS2 have_VH0 &lsqb;_( formula_NN1 &rsqb;_) In_II reverse_JJ order_NN1 ,_, we_PPIS2 find_VV0 &lsqb;_( formula_NN1 &rsqb;_) so_CS21 that_CS22 &lsqb;_( formula_NN1 &rsqb;_) and_CC hence_RR It_PPH1 follows_VVZ that_CST the_AT zeros_NN2 of_IO &lsqb;_( formula_NN1 &rsqb;_) also_RR make_VV0 &lsqb;_( formula_NN1 &rsqb;_) vanish_VV0 and_CC that_DD1 &lsqb;_( formula_NN1 &rsqb;_) must_VM also_RR have_VHI n_ZZ1 p_ZZ1 roots_NN2 &lsqb;_( formula_NN1 &rsqb;_) =_FO 0_MC ._. 
This_DD1 last_MD is_VBZ otherwise_RR obvious_JJ ;_; A_ZZ1 has_VHZ only_RR p_ZZ1 columns_NN2 ,_, so_CS21 that_CS22 AB_FO ,_, of_IO order_NN1 (_( n_ZZ1 n_ZZ1 )_) ,_, has_VHZ at_RR21 most_RR22 p_ZZ1 linearly_RR independent_JJ columns_NN2 ;_; i.e._REX it_PPH1 is_VBZ of_IO rank_NN1 p_ZZ1 at_RR21 most_RR22 ,_, and_CC so_RR must_VM have_VHI at_RR21 least_RR22 n_ZZ1 p_ZZ1 zero_MC eigenvalues_NN2 ._. 
If_CS A_ZZ1 and_CC B_ZZ1 are_VBR square_JJ ,_, and_CC one_PN1 at_RR21 least_RR22 (_( say_VV0 B_ZZ1 )_) is_VBZ non-singular_JJ ,_, this_DD1 result_NN1 may_VM be_VBI obtained_VVN very_RG simply_RR from_II the_AT identity._NNU &lsqb;_( formula_NN1 &rsqb;_) Corollary_NN1 If_CS A_ZZ1 ,_, B_ZZ1 an_AT1 C_ZZ1 are_VBR matrices_NN2 of_IO order_NN1 n_ZZ1 p_ZZ1 ,_, p_ZZ1 q_ZZ1 and_CC q_ZZ1 n_ZZ1 respectively_RR ,_, the_AT nonzero_NN1 eigenvalues_NN2 of_IO ABC_NP1 (_( n_ZZ1 n_ZZ1 )_) ,_, CAB_NN1 (_( q_ZZ1 q_ZZ1 )_) and_CC BCA_NP1 (_( p_ZZ1 p_ZZ1 )_) are_VBR identical_JJ ._. 
(_( N.B._FO if_CS n_ZZ1 =_FO p_ZZ1 =_FO q_ZZ1 ,_, this_DD1 corollary_NN1 applies_VVZ only_RR to_II the_AT products_NN2 ABC_NP1 ,_, CAB_NN1 and_CC BCA_NP1 and_CC not_XX to_II BAC_NP1 ,_, etc._RA ;_; i.e_REX cyclic_JJ permutation_NN1 only_RR is_VBZ permitted_VVN ._. )_) 
Example_NN1 VII_MC (_( a_ZZ1 )_) &lsqb;_( formula_NN1 &rsqb;_) Here_RL ,_, &lsqb;_( formula_NN1 &rsqb;_) The_AT reader_NN1 will_VM verify_VVI that_CST AB_FO is_VBZ of_IO rank_NN1 1_MC1 with_IW spectral_JJ and_CC modal_JJ matrices_NN2 &lsqb;_( formula_NN1 &rsqb;_) while_CS BA_NP1 is_VBZ of_IO rank_NN1 2_MC with_IW &lsqb;_( formula_NN1 &rsqb;_) The_AT eigenvalues_NN2 of_IO AB_FO and_CC BA_NP1 are_VBR thus_RR 0_MC ,_, 0_MC and_CC -7_MC ._. 
It_PPH1 will_VM be_VBI noted_VVN that_DD1 trBA_NN1 =_FO tr_JJ AB_FO =_FO -7_MC and_CC that_DD1 Theorem_NN1 VII_MC applies_VVZ regardless_RR of_IO the_AT fact_NN1 that_CST rank_NN1 &lsqb;_( formula_NN1 &rsqb;_) rank_NN1 BA._NN1 (_( b_ZZ1 )_) &lsqb;_( formula_NN1 &rsqb;_) Here_RL &lsqb;_( formula_NN1 &rsqb;_) each_DD1 product_NN1 being_VBG of_IO unit_NN1 rank_NN1 ._. 
The_AT reader_NN1 will_VM verify_VVI that_CST the_AT spectral_JJ and_CC modal_JJ matrices_NN2 are_VBR (_( i_ZZ1 )_) For_IF ABC_NP1 :_: &lsqb;_( formula_NN1 &rsqb;_) (_( ii_MC )_) For_IF CAB_NN1 :_: &lsqb;_( formula_NN1 &rsqb;_) (_( iii_MC )_) For_IF BCA_NP1 :_: &lsqb;_( formula_NN1 &rsqb;_) The_AT non-zero_JJ eigenvalue_NN1 ,_, 47_MC ,_, is_VBZ common_JJ to_II all_DB products_NN2 ,_, as_CSA indicated_VVN by_II Theorem_NN1 VII_MC ._. 
Theorem_NN1 VIII_MC If_CS ,_, in_II the_AT eigenvalue_NN1 problem_NN1 &lsqb;_( formula_NN1 &rsqb;_) the_AT matrices_NN2 B_ZZ1 ,_, C_ZZ1 are_VBR both_RR real_JJ and_CC symmetric_JJ ,_, and_CC if_CS further_RRR C_NP1 is_VBZ pos._NNU def._NNU and_CC B_ZZ1 non-neg._NNU def._NNU ,_, then_RT all_DB eigenvalues_NN2 are_VBR real_JJ ,_, finite_JJ and_CC non-negative_JJ ._. 
In_II (_( 8_MC )_) write_VV0 &lsqb;_( formula_NN1 &rsqb;_) ;_; multiply_VV0 out_RP and_CC separate_JJ real_JJ and_CC imaginary_JJ parts_NN2 ._. 
Two_MC real_JJ equations_NN2 result_VV0 :_: &lsqb;_( formula_NN1 &rsqb;_) If_CS these_DD2 equations_NN2 are_VBR respectively_RR premultiplied_VVN by_II &lsqb;_( formula_NN1 &rsqb;_) and_CC subtracted_VVD ,_, then_RT since_CS B_ZZ1 ,_, C_ZZ1 are_VBR symmetric_JJ ,_, the_AT left-hand_JJ side_NN1 vanishes_VVZ ,_, and_CC &lsqb;_( formula_NN1 &rsqb;_) Since_II x_ZZ1 is_VBZ not_XX null_JJ and_CC C_ZZ1 is_VBZ pos._NNU def._NNU ,_, the_AT bracketed_JJ term_NN1 is_VBZ positive_JJ ;_; hence_RR go_VV0 =_FO 0_MC and_CC &lsqb;_( formula_NN1 &rsqb;_) is_VBZ real_JJ ._. 
Equation_NN1 (_( 8_MC )_) is_VBZ now_RT &lsqb;_( formula_NN1 &rsqb;_) and_CC since_CS the_AT bracketed_JJ term_NN1 is_VBZ real_JJ ,_, so_RG also_RR is_VBZ x_ZZ1 ._. 
Finally_RR ,_, &lsqb;_( formula_NN1 &rsqb;_) and_CC with_IW the_AT given_JJ conditions_NN2 &lsqb;_( formula_NN1 &rsqb;_) is_VBZ therefore_RR real_JJ ,_, finite_JJ ,_, and_CC non-negative_JJ ._. 
I_ZZ1 B_ZZ1 ,_, C_ZZ1 are_VBR both_RR pos._NNU def._NNU ,_, &lsqb;_( formula_NN1 &rsqb;_) is_VBZ real_JJ ,_, finite_JJ ,_, an_AT1 positive_JJ ._. 
Corollaries_NN2 (_( i_ZZ1 )_) The_AT eigenvalues_NN2 of_IO a_AT1 real_JJ ,_, symmetric_JJ matrix_NN1 A_ZZ1 are_VBR all_DB real_JJ and_CC finite_JJ ,_, and_CC the_AT eigenvectors_NN2 are_VBR real_JJ ._. 
This_DD1 follows_VVZ from_II the_AT theorem_NN1 by_II putting_VVG B_ZZ1 =_FO A_ZZ1 ,_, C_ZZ1 =_FO I._NP1 (_( ii_MC )_) The_AT eigenvalues_NN2 of_IO a_AT1 Hermitian_JJ matrix_NN1 are_VBR all_DB real_JJ ._. 
The_AT relevant_JJ equation_NN1 may_VM be_VBI written_VVN &lsqb;_( formula_NN1 &rsqb;_) which_DDQ yields_VVZ the_AT two_MC real_JJ equations_NN2 &lsqb;_( formula_NN1 &rsqb;_) Premultiply_RR the_AT first_MD by_II qT_NNU and_CC the_AT second_NNT1 by_II pT_NNU ._. 
Since_CS B_ZZ1 is_VBZ symmetric_JJ ,_, the_AT left-hand_JJ sides_NN2 are_VBR equal_JJ ,_, an_AT1 since_CS C_ZZ1 is_VBZ skew-symmetric_JJ ,_, its_APPGE quadratic_JJ form_NN1 vanishes_VVZ (_( by_II transposition_NN1 &lsqb;_( formula_NN1 &rsqb;_) )_) ._. 
Subtraction_NN1 of_IO the_AT two_MC equations_NN2 therefore_RR yields_VVZ &lsqb;_( formula_NN1 &rsqb;_) (_( iii_MC )_) The_AT non-zero_JJ eigenvalues_NN2 of_IO a_AT1 skew-Hermitian_JJ matrix_NN1 are_VBR all_DB imaginary_JJ ._. 
Here_RL we_PPIS2 premultiply_RR the_AT first_MD of_IO Equations_NN2 (_( 10_MC )_) by_II &lsqb;_( formula_NN1 &rsqb;_) and_CC the_AT second_NNT1 by_II &lsqb;_( formula_NN1 &rsqb;_) ._. 
Now_RT ,_, since_CS B_ZZ1 is_VBZ skew-symmetric_JJ it_PPH1 disappears_VVZ from_II the_AT equations_NN2 ,_, while_CS C_ZZ1 is_VBZ now_RT symmetric_JJ ,_, so_CS21 that_CS22 the_AT right-hand_JJ sides_NN2 are_VBR equal_JJ and_CC opposite_JJ ._. 
Addition_NN1 then_RT gives_VVZ &lsqb;_( formula_NN1 &rsqb;_) Theorem_NN1 IX_MC If_CS a_AT1 matrix_NN1 A_ZZ1 and_CC its_APPGE inverse_NN1 A-1_MC1 are_VBR known_VVN ,_, then_RT if_CS a_AT1 unit_NN1 rank_NN1 matrix_NN1 is_VBZ added_VVN to_II A_ZZ1 ,_, the_AT inverse_NN1 of_IO the_AT new_JJ matrix_NN1 may_VM be_VBI written_VVN down_RP ._. 
Let_VV0 us_PPIO2 write_VVI the_AT unit_NN1 rank_NN1 matrix_NN1 as_CSA &lsqb;_( formula_NN1 &rsqb;_) Then_RT &lsqb;_( formula_NN1 &rsqb;_) Hence_RR &lsqb;_( formula_NN1 &rsqb;_) To_TO prove_VVI this_DD1 ,_, we_PPIS2 merely_RR multiply_VV0 out_RP the_AT product_NN1 &lsqb;_( formula_NN1 &rsqb;_) which_DDQ gives_VVZ I._NP1 Equation_NN1 (_( 11_MC )_) is_VBZ an_AT1 example_NN1 of_IO the_AT &quot;_" modification_NN1 formula_NN1 &quot;_" generally_RR ascribed_VVN to_II Householder_NP1 (_( 3_MC )_) ._. 
(_( Householder_NP1 's_GE formula_NN1 relates_VVZ to_II the_AT more_RGR general_JJ case_NN1 where_CS a_AT1 matrix_NN1 of_IO rank_NN1 &lsqb;_( formula_NN1 &rsqb;_) is_VBZ added_VVN to_II A._NP1 The_AT additional_JJ ,_, modifying_VVG ,_, matrix_NN1 may_VM be_VBI written_VVN CRT_NP1 with_IW C_ZZ1 of_IO order_NN1 (_( n_ZZ1 p_ZZ1 )_) and_CC RT_NN1 of_IO order_NN1 (_( p_ZZ1 n_ZZ1 )_) ._. 
Then_RT with_IW &lsqb;_( formula_NN1 &rsqb;_) an_AT1 analogy_NN1 with_IW Equation_NN1 (_( 11_MC )_) ._. 
The_AT reader_NN1 might_VM verify_VVI this_DD1 useful_JJ formula_NN1 in_II which_DDQ it_PPH1 is_VBZ evident_JJ that_CST the_AT inverse_NN1 of_IO a_AT1 matrix_NN1 of_IO order_NN1 n_ZZ1 is_VBZ obtained_VVN in_II31 terms_II32 of_II33 that_DD1 of_IO a_AT1 matrix_NN1 of_IO smaller_JJR order_NN1 )_) ._. 
Examples_NN2 of_IO the_AT use_NN1 of_IO this_DD1 theorem_NN1 are_VBR given_VVN in_II 2.2.3_MC ._. 
Theorem_NN1 X_ZZ1 If_CS A_ZZ1 is_VBZ real_JJ and_CC symmetric_JJ ,_, it_PPH1 can_VM not_XX be_VBI defective_JJ ._. 
For_IF suppose_VV0 it_PPH1 is_VBZ ;_; let_VV0 us_PPIO2 write_VVI its_APPGE spectral_JJ matrix_NN1 &lsqb;_( formula_NN1 &rsqb;_) with_IW a_AT1 Jordan_NP1 block_NN1 &lsqb;_( formula_NN1 &rsqb;_) as_II the_AT leading_JJ submatrix_NN1 of_IO order_NN1 2_MC (_( it_PPH1 can_VM be_VBI part_NN1 of_IO a_AT1 larger_JJR block_NN1 )_) ;_; x1_FO is_VBZ the_AT corresponding_JJ eigenvector_NN1 and_CC &lsqb;_( formula_NN1 &rsqb;_) the_AT auxiliary_JJ vector_NN1 ._. 
Then_RT from_II the_AT first_MD two_MC columns_NN2 of_IO &lsqb;_( formula_NN1 &rsqb;_) we_PPIS2 have_VH0 &lsqb;_( formula_NN1 &rsqb;_) on_II transposition_NN1 ,_, since_CS A_ZZ1 is_VBZ symmetric_JJ ;_; and_CC &lsqb;_( formula_NN1 &rsqb;_) Premultiply_NP1 (_( 13_MC )_) by_II &lsqb;_( formula_NN1 &rsqb;_) ;_; then_RT in_II31 view_II32 of_II33 (_( 12_MC )_) &lsqb;_( formula_NN1 &rsqb;_) In_II corollary_NN1 (_( i_ZZ1 )_) of_IO Theorem_NN1 VII_MC we_PPIS2 have_VH0 shown_VVN that_CST if_CS A_ZZ1 is_VBZ real_JJ and_CC symmetric_JJ ,_, x_ZZ1 is_VBZ real_JJ ;_; hence_RR in_II (_( 14_MC )_) &lsqb;_( formula_NN1 &rsqb;_) and_CC p_ZZ1 vanishes_VVZ ;_; similarly_RR for_IF all_DB other_JJ superdiagonal_JJ elements_NN2 ._. 
Hence_RR A_ZZ1 is_VBZ not_XX defective_JJ ._. 
Corollaries_NN2 (_( i_ZZ1 )_) If_CS A_ZZ1 is_VBZ not_XX symmetric_JJ ,_, but_CCB can_VM be_VBI written_VVN as_II the_AT product_NN1 &lsqb;_( formula_NN1 &rsqb;_) ,_, where_CS B_ZZ1 ,_, C_ZZ1 are_VBR real_JJ ad_NN1 symmetric_JJ ,_, and_CC in_RR21 addition_RR22 C_ZZ1 is_VBZ pos._NNU def._NNU ,_, then_RT A_ZZ1 can_VM not_XX be_VBI defective_JJ ._. 
In_II this_DD1 case_NN1 ,_, (_( 12_MC )_) becomes_VVZ successively_RR &lsqb;_( formula_NN1 &rsqb;_) and_CC (_( 13_MC )_) yields_NN2 &lsqb;_( formula_NN1 &rsqb;_) Hence_RR as_CSA before_RT ,_, &lsqb;_( formula_NN1 &rsqb;_) We_PPIS2 have_VH0 proved_VVN in_II Theorem_NN1 VIII_MC that_DD1 with_IW these_DD2 conditions_NN2 ,_, X_ZZ1 is_VBZ real_JJ ;_; hence_RR (_( 15_MC )_) implies_VVZ p_ZZ1 =_FO 0_MC ,_, and_CC as_CSA before_RT ,_, A_ZZ1 is_VBZ not_XX defective_JJ ._. 
Provided_CS only_RR that_CST A_ZZ1 is_VBZ real_JJ ,_, it_PPH1 is_VBZ always_RR possible_JJ to_TO find_VVI real_JJ symmetric_JJ matrices_NN2 B_ZZ1 ,_, C_ZZ1 satisfying_JJ CA_NP1 =_FO B_ZZ1 (_( see_VV0 2.10.3_MC )_) ._. 
Accordingly_RR ,_, if_CS A_ZZ1 is_VBZ defective_JJ ,_, so_CS21 that_CS22 &lsqb;_( formula_NN1 &rsqb;_) ,_, (_( 15_MC )_) requires_VVZ that_CST &lsqb;_( formula_NN1 &rsqb;_) must_VM vanish._NNU (_( ii_MC )_) A_ZZ1 defective_JJ matrix_NN1 can_VM not_XX be_VBI similarly_RR transformed_VVN into_II a_AT1 real_JJ symmetric_JJ matrix_NN1 ._. 
For_IF suppose_VV0 we_PPIS2 have_VH0 a_AT1 Jordan_NP1 block_NN1 as_CSA in_II the_AT theorem_NN1 ,_, with_IW &lsqb;_( formula_NN1 &rsqb;_) ,_, and_CC we_PPIS2 apply_VV0 the_AT most_RGT general_JJ transformation_NN1 ;_; we_PPIS2 obtain_VV0 &lsqb;_( formula_NN1 &rsqb;_) This_DD1 can_VM not_XX be_VBI real_JJ and_CC symmetric_JJ :_: it_PPH1 requires_VVZ c_ZZ1 =_FO d_ZZ1 =_FO 0_MC ,_, which_DDQ violates_VVZ the_AT condition_NN1 &lsqb;_( formula_NN1 &rsqb;_) ._. 
Example_NN1 X_ZZ1 We_PPIS2 now_RT illustrate_VV0 Equation_NN1 (_( 15_MC )_) ._. 
(_( a_ZZ1 )_) No-defective_JJ matrix_NN1 we_PPIS2 choose_VV0 the_AT matrix_NN1 defined_VVN in_II (_( 1.21.1_MC )_) ,_, and_CC write_VV0 it_PPH1 here_RL as_CSA A1_FO ._. 
Then_RT with_IW &lsqb;_( formula_NN1 &rsqb;_) ,_, one_MC1 possible_JJ numerical_JJ expression_NN1 is_VBZ &lsqb;_( formula_NN1 &rsqb;_) Here_RL B_ZZ1 ,_, C_ZZ1 are_VBR real_JJ and_CC symmetric_JJ ,_, and_CC C_ZZ1 is_VBZ pos._NNU def._NNU ;_; A1_FO has_VHZ two_MC equal_JJ eigenvalues_NN2 ,_, 2_MC ,_, and_CC two_MC eigenvectors_NN2 1,2,4_MC and_CC 1,0_MC ,_, -1_MC ._. 
These_DD2 give_VV0 &lsqb;_( formula_NN1 &rsqb;_) ._. 
With_IW p_ZZ1 0_MC these_DD2 satisfy_VV0 (_( 15_MC )_) ._. 
(_( b_ZZ1 )_) Defective_JJ matrix_NN1 we_PPIS2 use_VV0 the_AT matrix_NN1 defined_VVN in_II (_( 1.21.2_MC )_) ,_, writing_VVG it_PPH1 here_RL as_CSA &lsqb;_( formula_NN1 &rsqb;_) ._. 
A_AT1 possible_JJ choice_NN1 for_IF C_ZZ1 ,_, B_ZZ1 now_RT yields_VVZ &lsqb;_( formula_NN1 &rsqb;_) As_CSA in_II (_( a_ZZ1 )_) ,_, &lsqb;_( formula_NN1 &rsqb;_) has_VHZ two_MC equal_JJ eigenvalues_NN2 ,_, 2_MC ,_, but_CCB here_RL possesses_VVZ only_RR the_AT one_MC1 eigenvector_NN1 ,_, 1,2,4_MC ._. 
With_IW this_DD1 vector_NN1 &lsqb;_( formula_NN1 &rsqb;_) ._. 
This_DD1 permits_VVZ &lsqb;_( formula_NN1 &rsqb;_) in_II (_( 15_MC )_) ._. 
Theorem_NN1 XI_NN1 The_AT eigenvalues_NN2 and_CC vectors_NN2 of_IO a_AT1 complex_JJ matrix_NN1 may_VM be_VBI obtained_VVN by_II treating_VVG a_AT1 related_JJ real_JJ matrix_NN1 ._. 
Suppose_VV0 &lsqb;_( formula_NN1 &rsqb;_) where_RRQ &lsqb;_( formula_NN1 &rsqb;_) are_VBR all_DB complex_JJ ._. 
There_EX will_VM be_VBI a_AT1 conjugate_NN1 relation_NN1 &lsqb;_( formula_NN1 &rsqb;_) Write_VV0 A_ZZ1 =_FO B_ZZ1 +_FO iC_JJ ,_, with_IW B_ZZ1 ,_, C_ZZ1 real_JJ ;_; then_RT &lsqb;_( formula_NN1 &rsqb;_) These_DD2 equations_NN2 are_VBR contained_VVN ,_, twice_RR ,_, in_II the_AT single_JJ equation_NN1 &lsqb;_( formula_NN1 &rsqb;_) Hence_RR the_AT real_JJ matrix_NN1 &lsqb;_( formula_NN1 &rsqb;_) has_VHZ the_AT eigenvalues_NN2 &lsqb;_( formula_NN1 &rsqb;_) of_IO (_( 16_MC )_) ,_, and_CC the_AT corresponding_JJ eigenvectors_NN2 &lsqb;_( formula_NN1 &rsqb;_) Example_NN1 XI_NN1 Let_VV0 &lsqb;_( formula_NN1 &rsqb;_) Then_RT the_AT numerical_JJ form_NN1 of_IO (_( 12_MC )_) is_VBZ &lsqb;_( formula_NN1 &rsqb;_) so_CS21 that_CS22 the_AT eigenvalues_NN2 and_CC vectors_NN2 of_IO A_ZZ1 are_VBR 1_MC1 i_ZZ1 ,_, -1,2_MC +_FO i_ZZ1 and_CC 2_MC +_FO 3i_FO ,_, -1,1_MC -i_MC1 ._. 
In_II the_AT alternative_NN1 form_VV0 the_AT system_NN1 matrix_NN1 ,_, modal_JJ matrix_NN1 and_CC spectral_JJ matrix_NN1 of_IO (_( 17_MC )_) are_VBR ,_, respectively_RR ,_, &lsqb;_( formula_NN1 &rsqb;_) and_CC the_AT reader_NN1 is_VBZ invited_VVN to_TO check_VVI that_CST these_DD2 matrices_NN2 satisfy_VV0 (_( 17_MC )_) ._. 
Theorem_NN1 XII_MC Sylvester_NP1 's_GE Law_NN1 of_IO Degeneracy_NN1 :_: the_AT degeneracy_NN1 of_IO the_AT product_NN1 of_IO two_MC square_JJ matrices_NN2 is_VBZ at_RR21 least_RR22 as_RG great_JJ as_CSA the_AT degeneracy_NN1 of_IO either_DD1 factor_NN1 ,_, and_CC at_RR21 most_RR22 as_RG great_JJ as_CSA the_AT sum_NN1 of_IO the_AT degeneracies_NN2 of_IO the_AT factors_NN2 ,_, or_CC the_AT order_NN1 of_IO the_AT matrices_NN2 ,_, whichever_DDQV is_VBZ less_RRR ._. 
We_PPIS2 consider_VV0 the_AT product_NN1 C_ZZ1 =_FO AB_FO ,_, where_CS A_ZZ1 ,_, B_ZZ1 have_VH0 degeneracies_NN2 p_ZZ1 ,_, q_ZZ1 respectively_RR ._. 
Without_IW loss_NN1 of_IO generality_NN1 we_PPIS2 may_VM take_VVI &lsqb;_( formula_NN1 &rsqb;_) ;_; for_IF ,_, since_CS the_AT rank_NN1 of_IO a_AT1 matrix_NN1 is_VBZ unaltered_JJ by_II transposition_NN1 ,_, we_PPIS2 may_VM consider_VVI the_AT product_NN1 either_RR as_CSA C_ZZ1 =_FO AB_FO or_CC CT_NN1 =_FO BTAT_NP1 ,_, with_IW the_AT matrix_NN1 of_IO greater_JJR degeneracy_NN1 on_II the_AT right_NN1 ._. 
We_PPIS2 also_RR recall_VV0 the_AT result_NN1 proved_VVN in_II 1.12_MC :_: if_CS a_AT1 matrix_NN1 of_IO rank_NN1 r_ZZ1 is_VBZ multiplied_VVN by_II a_AT1 non-singular_JJ matrix_NN1 ,_, the_AT product_NN1 has_VHZ rank_NN1 r_ZZ1 ._. 
This_DD1 is_VBZ because_CS the_AT non-singular_JJ matrix_NN1 may_VM be_VBI regarded_VVN as_II made_VVD up_RP of_IO elementary_JJ operations_NN2 ,_, and_CC these_DD2 can_VM not_XX change_VVI the_AT rank_NN1 of_IO a_AT1 non-vanishing_JJ minor_NN1 of_IO order_NN1 r_ZZ1 ._. 
This_DD1 is_VBZ a_AT1 special_JJ case_NN1 of_IO Sylvester_NP1 's_GE law_NN1 :_: the_AT degeneracy_NN1 is_VBZ at_RR21 least_RR22 n_ZZ1 r_ZZ1 and_CC at_RR21 most_RR22 n_ZZ1 r_ZZ1 ;_; i.e._REX it_PPH1 is_VBZ n_ZZ1 r_ZZ1 ._. 
We_PPIS2 now_RT turn_VV0 to_II the_AT general_JJ case_NN1 ,_, and_CC approach_VV0 it_PPH1 by_II31 means_II32 of_II33 a_AT1 simple_JJ example_NN1 ._. 
Let_VV0 A_ZZ1 ,_, B_ZZ1 be_VBI standard_JJ canonical_JJ forms_NN2 ,_, with_IW n_ZZ1 p_ZZ1 ,_, n_ZZ1 q_ZZ1 consecutive_JJ units_NN2 in_II the_AT respective_JJ diagonals_NN2 ,_, beginning_VVG at_II top_JJ left_JJ ,_, and_CC with_IW zeros_NN2 elsewhere_RL ._. 
Since_CS &lsqb;_( formula_NN1 &rsqb;_) ,_, only_RR the_AT first_MD n_ZZ1 q_ZZ1 units_NN2 in_II A_ZZ1 are_VBR multiplied_VVN by_II units_NN2 in_II B_ZZ1 ,_, ie.e._NNU the_AT product_NN1 is_VBZ identical_JJ with_IW B_ZZ1 and_CC is_VBZ of_IO degeneracy_NN1 q_ZZ1 ._. 
Now_RT move_VV0 the_AT units_NN2 in_II A_ZZ1 one_MC1 place_NN1 down_II the_AT diagonal_JJ ._. 
The_AT leading_JJ element_NN1 in_II the_AT product_NN1 now_RT vanishes_VVZ ,_, i.e._REX the_AT degeneracy_NN1 is_VBZ now_RT q_ZZ1 +_FO 1_MC1 ._. 
Proceeding_VVG in_II this_DD1 way_NN1 ,_, we_PPIS2 finally_RR have_VH0 all_DB the_AT units_NN2 in_II A_ZZ1 at_II the_AT lower_JJR end_NN1 of_IO the_AT diagonal_JJ ,_, with_IW p_ZZ1 zeros_NN2 at_II the_AT top_NN1 ,_, so_CS21 that_CS22 the_AT degeneracy_NN1 of_IO the_AT product_NN1 is_VBZ now_RT p_ZZ1 +_FO q_ZZ1 (_( or_CC ,_, if_CS &lsqb;_( formula_NN1 &rsqb;_) ,_, the_AT product_NN1 is_VBZ null_JJ )_) ._. 
This_DD1 example_NN1 not_XX only_RR illustrates_VVZ Sylvester_NP1 's_GE law_NN1 ;_; it_PPH1 also_RR makes_VVZ the_AT important_JJ point_NN1 that_CST the_AT degeneracy_NN1 of_IO the_AT product_NN1 depends_VVZ on_II the_AT relative_JJ positions_NN2 of_IO the_AT units_NN2 ,_, and_CC therefore_RR of_IO the_AT linearly_RR independent_JJ columns_NN2 ,_, in_II the_AT two_MC factors_NN2 ._. 
Now_RT let_VV0 A_ZZ1 ,_, B_ZZ1 be_VBI general_JJ ,_, and_CC let_VV0 the_AT equivalent_JJ transformations_NN2 which_DDQ give_VV0 them_PPHO2 canonical_JJ form_NN1 be_VBI PAQ_JJ ,_, RBS_NP1 ,_, where_CS P_ZZ1 ,_, Q_ZZ1 ,_, R_ZZ1 ,_, S_ZZ1 are_VBR non-singular_JJ ._. 
Then_RT we_PPIS2 may_VM write_VVI &lsqb;_( formula_NN1 &rsqb;_) and_CC D_ZZ1 has_VHZ the_AT same_DA degeneracy_NN1 as_CSA C._NP1 The_AT matrix_NN1 &lsqb;_( formula_NN1 &rsqb;_) will_VM in_RR21 general_RR22 be_VBI fully_RR populated_VVN ;_; if_CS it_PPH1 is_VBZ appropriately_RR partitioned_VVN ,_, we_PPIS2 may_VM write_VVI (_( 18_MC )_) as_CSA &lsqb;_( formula_NN1 &rsqb;_) To_TO be_VBI conformable_JJ with_IW the_AT first_MD and_CC last_MD matrices_NN2 in_II the_AT triple_JJ product_NN1 ,_, &lsqb;_( formula_NN1 &rsqb;_) must_VM have_VHI p_NN1 rows_NN2 and_CC n_ZZ1 q_ZZ1 columns_NN2 ._. 
The_AT final_JJ matrix_NN1 then_RT shows_VVZ that_CST D_ZZ1 has_VHZ p_NNU null_JJ rows_NN2 and_CC q_ZZ1 null_JJ columns_NN2 ;_; since_CS &lsqb;_( formula_NN1 &rsqb;_) ,_, D_ZZ1 has_VHZ degeneracy_NN1 q_ZZ1 at_RR21 least_RR22 ._. 
This_DD1 establishes_VVZ the_AT first_MD part_NN1 of_IO the_AT theorem_NN1 ._. 
Whether_CSW31 or_CSW32 not_CSW33 D_ZZ1 has_VHZ greater_JJR degeneracy_NN1 than_CSN q_ZZ1 clearly_RR depends_VVZ on_II the_AT number_NN1 of_IO linearly_RR independent_JJ columns_NN2 in_II &lsqb;_( formula_NN1 &rsqb;_) To_TO examine_VVI this_DD1 question_NN1 ,_, consider_VV0 the_AT submatrix_NN1 &lsqb;_( formula_NN1 &rsqb;_) The_AT n_ZZ1 p_ZZ1 rows_NN2 of_IO this_DD1 submatrix_NN1 ,_, being_VBG part_NN1 of_IO &lsqb;_( formula_NN1 &rsqb;_) ,_, a_AT1 on-singular_JJ matrix_NN1 ,_, are_VBR necessarily_RR linearly_RR independent_JJ ._. 
In_II consequence_NN1 ,_, &lsqb;_( formula_NN1 &rsqb;_) must_VM have_VHI just_RR n_ZZ1 p_ZZ1 linearly_RR independent_JJ columns_NN2 ,_, making_VVG up_RP a_AT1 non-vanishing_JJ minor_NN1 of_IO order_NN1 n_ZZ1 p_ZZ1 ;_; the_AT other_JJ p_NN1 columns_NN2 are_VBR either_RR linear_JJ combinations_NN2 of_IO some_DD or_CC all_DB of_IO the_AT columns_NN2 of_IO the_AT minor_JJ ,_, or_CC null_JJ (_( a_AT1 special_JJ combination_NN1 with_IW all_DB coefficients_NN2 zero_MC )_) ._. 
First_MD ,_, suppose_VV0 &lsqb;_( formula_NN1 &rsqb;_) ._. 
Now_RT &lsqb;_( formula_NN1 &rsqb;_) has_VHZ n_ZZ1 q_ZZ1 columns_NN2 ,_, the_AT minor_JJ n_ZZ1 p_ZZ1 columns_NN2 :_: separately_RR ,_, a_AT1 total_NN1 of_IO 2n_FO (_( p_ZZ1 +_FO q_ZZ1 )_) ._. 
But_CCB &lsqb;_( formula_NN1 &rsqb;_) has_VHZ only_RR n_ZZ1 columns_NN2 ;_; hence_RR &lsqb;_( formula_NN1 &rsqb;_) and_CC the_AT minor_JJ have_VH0 at_RR21 least_RR22 n_ZZ1 (_( p_ZZ1 +_FO q_ZZ1 )_) common_JJ columns_NN2 ,_, i.e._REX &lsqb;_( formula_NN1 &rsqb;_) has_VHZ at_RR21 least_RR22 n_ZZ1 (_( p_ZZ1 +_FO q_ZZ1 )_) linearly_RR independent_JJ columns_NN2 ._. 
The_AT remaining_JJ p_NN1 columns_NN2 of_IO &lsqb;_( formula_NN1 &rsqb;_) may_VM themselves_PPX2 be_VBI columns_NN2 of_IO the_AT minor_JJ ,_, or_CC may_VM be_VBI linear_JJ combinations_NN2 of_IO some_DD or_CC all_DB of_IO the_AT columns_NN2 of_IO the_AT minor_JJ ,_, or_CC may_VM be_VBI linear_JJ combinations_NN2 of_IO the_AT n_ZZ1 (_( p_ZZ1 +_FO q_ZZ1 )_) common_JJ columns_NN2 ,_, or_CC may_VM be_VBI null_JJ ._. 
At_II the_AT extremes_NN2 ,_, all_DB columns_NN2 of_IO &lsqb;_( formula_NN1 &rsqb;_) or_CC n_ZZ1 (_( p_ZZ1 +_FO q_ZZ1 )_) columns_NN2 of_IO ga_NN1 will_VM be_VBI linearly_RR independent_JJ ,_, giving_VVG D_ZZ1 degeneracies_NN2 of_IO q_ZZ1 and_CC p_ZZ1 +_FO q_ZZ1 respectively_RR ;_; any_DD intermediate_JJ degeneracy_NN1 is_VBZ clearly_RR possible_JJ ._. 
Finally_RR ,_, suppose_VV0 &lsqb;_( formula_NN1 &rsqb;_) ._. 
Then_RT ,_, as_CSA before_RT ,_, at_II one_MC1 extreme_JJ ,_, &lsqb;_( formula_NN1 &rsqb;_) can_VM contain_VVI n_ZZ1 q_ZZ1 columns_NN2 of_IO the_AT minor_JJ ,_, which_DDQ are_VBR linearly_RR independent_JJ ;_; at_II the_AT other_JJ extreme_JJ ,_, the_AT minor_NN1 could_VM be_VBI wholly_RR contained_VVN in_II &lsqb;_( formula_NN1 &rsqb;_) and_CC &lsqb;_( formula_NN1 &rsqb;_) could_VM be_VBI null_JJ ,_, when_CS D_ZZ1 has_VHZ degeneracy_NN1 &lsqb;_( formula_NN1 &rsqb;_) ._. 
This_DD1 establishes_VVZ the_AT second_MD part_NN1 of_IO Sylvester_NP1 's_GE theorem_NN1 ._. 
Examples_NN2 XII_MC Although_CS in_II a_AT1 numerical_JJ case_NN1 we_PPIS2 can_VM find_VVI the_AT degeneracy_NN1 of_IO AB_FO by_II evaluating_VVG and_CC studying_VVG C_ZZ1 ,_, it_PPH1 is_VBZ instructive_JJ to_TO examine_VVI (_( 18_MC )_) in_II numerical_JJ cases_NN2 ._. 
We_PPIS2 give_VV0 two_MC examples_NN2 (_( i_ZZ1 )_) Let_VV0 &lsqb;_( formula_NN1 &rsqb;_) Here_RL A_ZZ1 is_VBZ of_IO degeneracy_NN1 1_MC1 ,_, B_ZZ1 and_CC C_NP1 each_DD1 of_IO degeneracy_NN1 2_MC ._. 
Now_RT by_II building_VVG up_RP P_ZZ1 and_CC Q_ZZ1 by_II elementary_JJ operations_NN2 (_( see_VV0 1.12_MC )_) we_PPIS2 obtain_VV0 &lsqb;_( formula_NN1 &rsqb;_) giving_VVG the_AT canonical_JJ form_NN1 for_IF A._NP1 Similarly_RR ,_, for_IF B_ZZ1 we_PPIS2 obtain_VV0 &lsqb;_( formula_NN1 &rsqb;_) Inverting_VVG Q_ZZ1 and_CC R_ZZ1 ,_, we_PPIS2 find_VV0 &lsqb;_( formula_NN1 &rsqb;_) Hence_RR D_ZZ1 is_VBZ given_VVN by_II &lsqb;_( formula_NN1 &rsqb;_) and_CC D_ZZ1 is_VBZ of_IO degeneracy_NN1 2_MC ._. 
Note_VV0 that_CST here_RL the_AT submatrix_NN1 &lsqb;_( formula_NN1 &rsqb;_) is_VBZ &lsqb;_( formula_NN1 &rsqb;_) and_CC we_PPIS2 can_VM take_VVI the_AT first_MD two_MC columns_NN2 as_CSA linearly_RR independent_JJ ;_; the_AT third_MD is_VBZ obtained_VVN by_II subtracting_VVG half_DB the_AT first_MD from_II the_AT second._NNU (_( ii_MC )_) If_CS we_PPIS2 transfer_VV0 the_AT last_MD column_NN1 of_IO A_ZZ1 to_II the_AT first_MD position_NN1 ,_, we_PPIS2 have_VH0 a_AT1 new_JJ matrix_NN1 A_ZZ1 ;_; with_IW B_ZZ1 unchanged_JJ we_PPIS2 then_RT have_VH0 &lsqb;_( formula_NN1 &rsqb;_) A_ZZ1 is_VBZ again_RT of_IO degeneracy_NN1 1_MC1 ,_, B_ZZ1 of_IO degeneracy_NN1 2_MC ;_; but_CCB here_RL C_ZZ1 is_VBZ of_IO degeneracy_NN1 3_MC ._. 
We_PPIS2 leave_VV0 it_PPH1 to_II the_AT reader_NN1 to_TO follow_VVI the_AT steps_NN2 of_IO Example_NN1 (_( i_ZZ1 )_) ,_, merely_RR noting_VVG that_CST ,_, in_II the_AT present_JJ example_NN1 ,_, &lsqb;_( formula_NN1 &rsqb;_) is_VBZ &lsqb;_( formula_NN1 &rsqb;_) with_IW a_AT1 null_JJ column_NN1 in_II the_AT first_MD position_NN1 ._. 
Theorem_NN1 XIII_MC a_AT1 necessary_JJ and_CC sufficient_JJ condition_NN1 that_CST a_AT1 real_JJ symmetric_JJ matrix_NN1 A_ZZ1 shall_VM be_VBI positive_JJ definite_JJ is_VBZ that_CST all_DB its_APPGE leading_JJ discriminants_NN2 shall_VM be_VBI positive_JJ ._. 
If_CS &lsqb;_( formula_NN1 &rsqb;_) is_VBZ the_AT real_JJ symmetric_JJ matrix_NN1 of_IO a_AT1 quadratic_JJ form_NN1 &lsqb;_( formula_NN1 &rsqb;_) ,_, then_RT the_AT determinant_NN1 &lsqb;_( formula_NN1 &rsqb;_) is_VBZ called_VVN the_AT discriminant_JJ of_IO the_AT form_NN1 ._. 
The_AT leading_JJ minor_JJ discriminant_JJ of_IO order_NN1 1_MC1 is_VBZ &lsqb;_( formula_NN1 &rsqb;_) of_IO order_NN1 2_MC is_VBZ &lsqb;_( formula_NN1 &rsqb;_) ,_, and_RR31 so_RR32 on_RR33 ._. 
Thus_RR if_CS A_ZZ1 is_VBZ to_TO be_VBI pos._NNU def._NNU ,_, the_AT theorem_NN1 requires_VVZ that_CST &lsqb;_( formula_NN1 &rsqb;_) ,_, all_DB i_ZZ1 ._. 
The_AT vector_NN1 x_ZZ1 is_VBZ real_JJ and_CC non-null_JJ but_CCB otherwise_RR arbitrary_JJ ._. 
First_MD ,_, we_PPIS2 put_VV0 x_ZZ1 =_FO e1._FO then_RT &lsqb;_( formula_NN1 &rsqb;_) reduced_VVD to_II &lsqb;_( formula_NN1 &rsqb;_) ,_, which_DDQ must_VM be_VBI positive_JJ if_CS f_ZZ1 is_VBZ pos._NNU def_NN1 ._. 
Next_MD ,_, put_VVN &lsqb;_( formula_NN1 &rsqb;_) Now_RT f_ZZ1 reduces_VVZ to_II &lsqb;_( formula_NN1 &rsqb;_) which_DDQ ,_, with_IW &lsqb;_( formula_NN1 &rsqb;_) requires_VVZ also_RR that_DD1 &lsqb;_( formula_NN1 &rsqb;_) ._. 
We_PPIS2 might_VM proceed_VVI in_II this_DD1 way_NN1 ,_, but_CCB a_AT1 different_JJ approach_NN1 is_VBZ simpler_JJR ._. 
In_II the_AT full_JJ expansion_NN1 of_IO f_ZZ1 ,_, collect_VV0 together_RL all_DB the_AT terms_NN2 involving_VVG x1_FO ;_; they_PPHS2 may_VM be_VBI written_VVN as_II a_AT1 perfect_JJ square_NN1 ,_, so_CS21 that_CS22 &lsqb;_( formula_NN1 &rsqb;_) where_RRQ B_ZZ1 is_VBZ a_AT1 quadratic_JJ form_NN1 involving_VVG &lsqb;_( formula_NN1 &rsqb;_) ,_, ..._... ,_, xn_FO only_RR ._. 
Now_RT we_PPIS2 collect_VV0 all_DB terms_NN2 involving_VVG &lsqb;_( formula_NN1 &rsqb;_) and_CC write_VV0 them_PPHO2 as_II a_AT1 perfect_JJ square_NN1 ,_, and_RR31 so_RR32 on_RR33 ,_, so_CS21 that_CS22 ultimately_RR we_PPIS2 can_VM write_VVI the_AT form_NN1 as_CSA &lsqb;_( formula_NN1 &rsqb;_) where_CS in_II fact_NN1 &lsqb;_( formula_NN1 &rsqb;_) and_CC &lsqb;_( formula_NN1 &rsqb;_) while_CS y_ZZ1 is_VBZ related_VVN to_II x_ZZ1 by_II the_AT triangular_JJ substitution_NN1 (_( see_VV0 (_( 20_MC )_) )_) &lsqb;_( formula_NN1 &rsqb;_) and_CC &lsqb;_( formula_NN1 &rsqb;_) ._. 
Hence_RR f_ZZ1 =_FO &lsqb;_( formula_NN1 &rsqb;_) ,_, so_CS21 that_CS22 &lsqb;_( formula_NN1 &rsqb;_) ,_, and_CC &lsqb;_( formula_NN1 &rsqb;_) Now_RT put_VV0 xn_FO =_FO 0_MC ._. 
Then_RT f_ZZ1 may_VM now_RT be_VBI written_VVN as_II the_AT quadratic_JJ form_NN1 of_IO which_DDQ the_AT matrix_NN1 is_VBZ the_AT leading_JJ minor_NN1 of_IO A_ZZ1 of_IO order_NN1 b_ZZ1 1_MC1 ,_, and_CC we_PPIS2 may_VM deduce_VVI ,_, corresponding_VVG to_II (_( 22_MC )_) ,_, &lsqb;_( formula_NN1 &rsqb;_) Now_RT ,_, by_II putting_VVG &lsqb;_( formula_NN1 &rsqb;_) we_PPIS2 establish_VV0 a_AT1 similar_JJ result_NN1 for_IF &lsqb;_( formula_NN1 &rsqb;_) and_RR31 so_RR32 on_RR33 ,_, so_CS21 that_CS22 finally_RR we_PPIS2 establish_VV0 that_CST &lsqb;_( formula_NN1 &rsqb;_) From_II (_( 21_MC )_) it_PPH1 is_VBZ clear_JJ that_CST a_AT1 necessary_JJ and_CC sufficient_JJ condition_NN1 that_CST f_ZZ1 shall_VM be_VBI positive_JJ is_VBZ that_CST Di_NP1 shall_VM be_VBI positive_JJ ,_, all_DB i_ZZ1 ._. 
It_PPH1 then_RT follows_VVZ from_II (_( 23_MC )_) that_CST all_DB the_AT discriminants_NN2 &lsqb;_( formula_NN1 &rsqb;_) must_VM also_RR be_VBI positive_JJ ;_; this_DD1 also_RR is_VBZ both_RR necessary_JJ and_CC sufficient_JJ ._. 
This_DD1 theorem_NN1 was_VBDZ first_MD established_VVN by_II Sylvester_NP1 ._. 
Theorem_NN1 XIV_MC If_CS two_MC matrices_NN2 A_ZZ1 ,_, B_ZZ1 permute_VV0 ,_, then_RT provided_VVD at_RR21 least_RR22 one_PN1 is_VBZ non-defective_JJ ,_, they_PPHS2 share_VV0 the_AT same_DA modal_JJ matrix_NN1 ._. 
Let_VV0 &lsqb;_( formula_NN1 &rsqb;_) where_RRQ C_ZZ1 is_VBZ the_AT canonical_JJ spectral_JJ matrix_NN1 &lsqb;_( formula_NN1 &rsqb;_) of_IO A_ZZ1 ,_, and_CC X_ZZ1 is_VBZ the_AT modal_JJ matrix_NN1 of_IO A:X_FO is_VBZ not_XX unique_JJ ;_; if_CS we_PPIS2 have_VH0 any_DD matrix_NN1 Y_ZZ1 which_DDQ permutes_VVZ with_IW C_NP1 ,_, then_RT from_II (_( 24_MC )_) AXY_NP1 =_FO XCY_NP1 =_FO XYC_NN1 so_CS21 that_CS22 we_PPIS2 can_VM write_VVI the_AT modal_JJ matrix_NN1 of_IO A_ZZ1 alternatively_RR as_CSA XY_FO ._. 
Now_RT write_VV0 &lsqb;_( formula_NN1 &rsqb;_) so_CS21 that_CS22 D_ZZ1 is_VBZ a_AT1 similar_JJ transform_NN1 of_IO B._NP1 Then_RT it_PPH1 follows_VVZ from_II AB_FO =_FO BA_NP1 ,_, on_II use_NN1 of_IO (_( 24_MC )_) and_CC (_( 26_MC )_) ,_, that_DD1 CD_NN1 =_FO DC_NP1 ._. 
We_PPIS2 now_RT consider_VV0 three_MC cases._NNU (_( a_ZZ1 )_) The_AT eigenvalues_NN2 of_IO A_ZZ1 are_VBR all_DB different_JJ ._. 
Thus_RR A_ZZ1 is_VBZ non-defective_JJ ,_, and_CC C_ZZ1 is_VBZ diagonal_JJ ,_, all_DB the_AT diagonal_JJ elements_NN2 being_VBG different_JJ ._. 
In_II this_DD1 case_NN1 (_( see_VV0 1.9(3)_FO )_) ,_, D_ZZ1 must_VM also_RR be_VBI diagonal_JJ ;_; (_( 26_MC )_) then_RT shows_VVZ that_CST D_ZZ1 is_VBZ the_AT spectral_JJ matrix_NN1 of_IO B_ZZ1 ,_, which_DDQ thus_RR shares_VVZ the_AT modal_JJ matrix_NN1 X_ZZ1 with_IW A._NP1 Note_VV0 that_CST D_ZZ1 can_VM have_VHI equal_JJ elements_NN2 (_( including_II zeros_NN2 )_) in_II its_APPGE diagonal._NNU (_( b_ZZ1 )_) A_ZZ1 is_VBZ non-defective_JJ ,_, but_CCB has_VHZ some_DD repeated_VVN eigenvalues_NN2 ._. 
Thus_RR C_ZZ1 is_VBZ diagonal_JJ ;_; we_PPIS2 assume_VV0 it_PPH1 to_TO be_VBI in_II standard_JJ canonical_JJ form_NN1 ,_, i.e._REX it_PPH1 has_VHZ equal_JJ eigenvalues_NN2 grouped_VVN together_RL :_: this_DD1 only_RR requires_VVZ the_AT columns_NN2 of_IO X_ZZ1 to_TO be_VBI written_VVN in_II appropriate_JJ order_NN1 ._. 
Suppose_VV0 ,_, for_REX21 example_REX22 ,_, that_CST C_ZZ1 has_VHZ m_MC eigenvalues_NN2 ga_NN1 ,_, in_II a_AT1 submatrix_NN1 of_IO order_NN1 m_ZZ1 in_II the_AT leading_JJ diagonal_JJ place_NN1 :_: there_EX may_VM be_VBI similar_JJ submatrices_NN2 down_II the_AT diagonal_JJ ._. 
Thus_RR C_ZZ1 possesses_VVZ a_AT1 scalar_JJ submatrix_NN1 &lsqb;_( formula_NN1 &rsqb;_) in_II the_AT leading_JJ position_NN1 ;_; Equation_NN1 (_( 28_MC )_) (_( see_VV0 1.9(3)_FO again_RT )_) now_RT permits_VVZ D_NP1 to_TO have_VHI a_AT1 corresponding_JJ submatrix_NN1 ,_, also_RR of_IO order_NN1 m_ZZ1 ,_, arbitrarily_RR populated_VVN :_: thus_RR D_ZZ1 will_VM have_VHI a_AT1 block_NN1 diagonal_JJ form_NN1 ._. 
Now_RT in_II this_DD1 case_NN1 D_ZZ1 can_VM be_VBI reduced_VVN to_II a_AT1 standard_JJ canonical_JJ form_NN1 E_ZZ1 by_II a_AT1 similar_JJ transformation_NN1 &lsqb;_( formula_NN1 &rsqb;_) where_RRQ Y_ZZ1 has_VHZ precisely_RR the_AT same_DA block_NN1 diagonal_JJ form_NN1 as_CSA D_ZZ1 ;_; in_II effect_NN1 ,_, each_DD1 diagonal_JJ block_NN1 is_VBZ resolved_VVN into_II its_APPGE own_DA canonical_JJ form_NN1 ._. 
But_CCB in_II this_DD1 case_NN1 ,_, Y_ZZ1 will_VM ,_, like_II D_ZZ1 ,_, permute_VV0 with_IW C_NP1 ;_; thus_RR from_II (_( 25_MC )_) the_AT modal_JJ matrix_NN1 of_IO A_ZZ1 may_VM be_VBI written_VVN XY_FO ,_, while_CS (_( 26_MC )_) and_CC (_( 29_MC )_) require_VV0 &lsqb;_( formula_NN1 &rsqb;_) so_CS21 that_CS22 B_ZZ1 shares_VVZ the_AT modal_JJ matrix_NN1 XY_FO with_IW A._NP1 Note_VV0 that_CST here_RL ,_, E_ZZ1 need_VM not_XX be_VBI diagonal_JJ ,_, so_CS21 that_CS22 B_ZZ1 can_VM be_VBI defective_JJ ;_; but_CCB non-zero_JJ elements_NN2 in_II the_AT superdiagonal_JJ of_IO E_ZZ1 can_VM only_RR occur_VVI in_II a_AT1 submatrix_NN1 corresponding_VVG to_II a_AT1 scalar_JJ submatrix_NN1 in_II C._NP1 This_DD1 establishes_VVZ the_AT theorem_NN1 as_CSA stated._NNU (_( c_ZZ1 )_) For_IF completeness_NN1 ,_, we_PPIS2 look_VV0 briefly_RR at_II the_AT case_NN1 where_CS A_ZZ1 is_VBZ defective_JJ ._. 
Suppose_VV0 ,_, for_REX21 example_REX22 ,_, that_CST C_ZZ1 has_VHZ the_AT leading_JJ submatrix_NN1 &lsqb;_( formula_NN1 &rsqb;_) then_RT the_AT most_RGT general_JJ submatrix_NN1 of_IO D_ZZ1 which_DDQ will_VM permute_VVI with_IW C1_FO is_VBZ &lsqb;_( formula_NN1 &rsqb;_) where_RRQ d_ZZ1 ,_, e_ZZ1 ,_, f_ZZ1 are_VBR arbitrary_JJ ._. 
Thus_RR D1_FO is_VBZ in_RR21 general_RR22 not_XX of_IO standard_JJ canonical_JJ form_NN1 ;_; to_TO make_VVI it_PPH1 so_RR we_PPIS2 must_VM at_RR21 least_RR22 remove_VVI f_ZZ1 ._. 
Now_RT while_CS ,_, we_PPIS2 can_VM find_VVI a_AT1 submatrix_NN1 Y1_FO such_CS21 that_CS22 &lsqb;_( formula_NN1 &rsqb;_) where_RRQ E1_FO is_VBZ of_IO standard_JJ canonical_JJ form_NN1 ,_, Y1_FO is_VBZ not_XX of_IO the_AT general_JJ form_NN1 D1_FO and_CC so_RR will_VM not_XX permute_VVI with_IW C1_FO ._. 
In_II this_DD1 case_NN1 ,_, A_ZZ1 has_VHZ the_AT modal_JJ matrix_NN1 X_ZZ1 ,_, B_ZZ1 the_AT modal_JJ matrix_NN1 XY_FO ._. 
Of_RR21 course_RR22 ,_, it_PPH1 is_VBZ possible_JJ for_IF B_ZZ1 to_TO be_VBI such_CS21 that_CS22 f_ZZ1 =_FO 0_MC ,_, when_CS D1_FO is_VBZ in_II standard_JJ canonical_JJ form_NN1 :_: A_ZZ1 and_CC B_ZZ1 then_RT share_VV0 the_AT modal_JJ matrix_NN1 X_ZZ1 ,_, though_CS both_DB2 are_VBR defective_JJ ._. 
The_AT obvious_JJ (_( trivial_JJ )_) example_NN1 of_IO this_DD1 is_VBZ that_DD1 of_IO a_AT1 defective_JJ matrix_NN1 permuting_VVG with_IW itself_PPX1 ._. 
But_CCB in_RR21 general_RR22 ,_, if_CS A_ZZ1 and_CC B_ZZ1 are_VBR both_RR defective_JJ ,_, they_PPHS2 do_VD0 not_XX have_VHI a_AT1 common_JJ modal_JJ matrix_NN1 ._. 
In_II41 the_II42 light_II43 of_II44 the_AT above_JJ ,_, it_PPH1 is_VBZ clear_JJ that_CST two_MC matrices_NN2 having_VHG the_AT same_DA modal_JJ matrix_NN1 do_VD0 not_XX necessarily_RR permute_VVI ;_; they_PPHS2 will_VM do_VDI so_RG only_RR if_CS their_APPGE spectral_JJ matrices_NN2 permute_VV0 ._. 
Corollaries_NN2 (_( i_ZZ1 )_) If_CS A_ZZ1 ,_, B_ZZ1 permute_VV0 ,_, so_RR do_VD0 powers_NN2 of_IO A_ZZ1 ,_, B._NP1 For_REX21 example_REX22 &lsqb;_( formula_NN1 &rsqb;_) (_( ii_MC )_) If_CS A_ZZ1 ,_, B_ZZ1 permute_VV0 ,_, and_CC the_AT eigenvalues_NN2 of_IO A_ZZ1 are_VBR all_DB different_JJ ,_, then_RT B_ZZ1 can_VM be_VBI expressed_VVN as_II a_AT1 polynomial_NN1 in_II A._NP1 As_CSA in_CS21 case_CS22 (_( a_ZZ1 )_) above_RL ,_, here_RL &lsqb;_( formula_NN1 &rsqb;_) where_RRQ C_ZZ1 ,_, D_ZZ1 are_VBR both_RR diagonal_JJ ._. 
Now_RT consider_VV0 the_AT general_JJ polynomial_NN1 &lsqb;_( formula_NN1 &rsqb;_) We_PPIS2 can_VM identify_VVI P(C)_NN1 with_IW D_ZZ1 ;_; this_DD1 requires_VVZ &lsqb;_( formula_NN1 &rsqb;_) where_RRQ &lsqb;_( formula_NN1 &rsqb;_) 1_MC1 ,_, ..._... ,_, &lsqb;_( formula_NN1 &rsqb;_) n_ZZ1 are_VBR the_AT diagonal_JJ elements_NN2 of_IO C_NP1 ,_, and_CC d1_FO ,_, ..._... ,_, dn_NNU those_DD2 of_IO D._NP1 The_AT square_JJ matrix_NN1 in_II (_( 30_MC )_) is_VBZ known_VVN as_II an_AT1 alternant_NN1 :_: its_APPGE reciprocal_JJ is_VBZ discussed_VVN in_II Ref_NN1 (_( P1_FO )_) ._. 
Equation_NN1 (_( 30_MC )_) defines_VVZ the_AT coefficients_NN2 &lsqb;_( formula_NN1 &rsqb;_) such_CS21 that_CS22 &lsqb;_( formula_NN1 &rsqb;_) which_DDQ establishes_VVZ the_AT corollary_NN1 ._. 
It_PPH1 is_VBZ capable_JJ of_IO extension_NN1 ,_, but_CCB we_PPIS2 shall_VM not_XX pursue_VVI the_AT matter_NN1 here_RL ._. 
There_EX is_VBZ a_AT1 considerable_JJ literature_NN1 concerning_II permutable_JJ matrices_NN2 ,_, see_VV0 ,_, for_REX21 example_REX22 ,_, Turnbull_NP1 an_AT1 Aitken_NN1 (_( 4_MC )_) ._. 
Theorem_NN1 XV_MC Sylvester_NP1 's_GE &quot;_" Law_NN1 of_IO Inertia_NN1 &quot;_" Let_VV0 a_AT1 real_JJ symmetric_JJ matrix_NN1 A_ZZ1 be_VBI rendered_VVN diagonal_JJ by_II a_AT1 congruent_JJ transformation_NN1 &lsqb;_( formula_NN1 &rsqb;_) ,_, where_CS B_ZZ1 is_VBZ ,_, of_RR21 course_RR22 ,_, non-singular_NN1 ._. 
Then_RT the_AT numbers_NN2 of_IO positive_JJ ,_, zero_MC ,_, and_CC negative_JJ elements_NN2 in_II the_AT diagonal_JJ of_IO &lsqb;_( formula_NN1 &rsqb;_) are_VBR invariant_JJ with_IW B._NP1 The_AT number_NN1 of_IO matrices_NN2 B_ZZ1 which_DDQ diagonalise_VV0 A_ZZ1 in_II this_DD1 manner_NN1 is_VBZ indefinitely_RR large_JJ :_: to_TO prove_VVI the_AT theorem_NN1 we_PPIS2 need_VV0 only_RR show_VVI that_CST it_PPH1 holds_VVZ for_IF any_DD two_MC ._. 
Our_APPGE first_MD choice_NN1 is_VBZ B_ZZ1 =_FO X_ZZ1 ,_, where_CS X_ZZ1 is_VBZ the_AT orthogonal_JJ modal_JJ matrix_NN1 of_IO A_ZZ1 (_( being_VBG symmetrical_JJ ,_, A_ZZ1 can_VM not_XX be_VBI defective_JJ :_: see_VV0 Theorem_NN1 X_ZZ1 )_) so_CS21 that_CS22 &lsqb;_( formula_NN1 &rsqb;_) ,_, the_AT spectral_JJ matrix_NN1 of_IO A._NP1 Our_APPGE other_JJ choice_NN1 is_VBZ &lsqb;_( formula_NN1 &rsqb;_) ,_, where_CS L_ZZ1 is_VBZ the_AT lower_JJR triangular_JJ matrix_NN1 having_VHG units_NN2 in_II its_APPGE diagonal_JJ and_CC its_APPGE other_JJ elements_NN2 so_RR chosen_VVN that_CST LA_NP1 is_VBZ upper_JJ triangular_JJ ,_, when_RRQ &lsqb;_( formula_NN1 &rsqb;_) ,_, a_AT1 diagonal_JJ matrix_NN1 ._. 
It_PPH1 is_VBZ a_AT1 simple_JJ exercise_NN1 ,_, which_DDQ we_PPIS2 leave_VV0 to_II the_AT reader_NN1 ,_, to_TO show_VVI that_CST the_AT process_NN1 of_IO triangulation_NN1 of_IO A_ZZ1 by_II LA_NP1 without_IW row_NN1 interchanges_NN2 gives_VVZ the_AT elements_NN2 &lsqb;_( formula_NN1 &rsqb;_) discussed_VVD in_II Theorem_NN1 XIII_MC ,_, where_CS the_AT &lsqb;_( formula_NN1 &rsqb;_) are_VBR the_AT principal_JJ discriminants_NN2 of_IO A_ZZ1 and_CC &lsqb;_( formula_NN1 &rsqb;_) :_: and_CC example_NN1 (_( for_IF a_AT1 non-symmetric_JJ matrix_NN1 )_) is_VBZ given_VVN in_II &lsqb;_( formula_NN1 &rsqb;_) We_PPIS2 first_MD note_NN1 that_CST if_CS A_ZZ1 is_VBZ of_IO rank_NN1 r_ZZ1 ,_, then_RT so_RR are_VBR &lsqb;_( formula_NN1 &rsqb;_) ;_; for_IF since_RR &lsqb;_( formula_NN1 &rsqb;_) ,_, B_ZZ1 ,_, may_VM be_VBI regarded_VVN as_II made_VVD of_IO elementary_JJ operations_NN2 &lsqb;_( formula_NN1 &rsqb;_) which_DDQ can_VM not_XX change_VVI the_AT rank_NN1 of_IO a_AT1 matrix_NN1 ._. 
Thus_RR both_RR &lsqb;_( formula_NN1 &rsqb;_) and_CC D_ZZ1 will_VM have_VHI n_ZZ1 r_ZZ1 zeros_NN2 in_II their_APPGE diagonals_NN2 ,_, which_DDQ establishes_VVZ part_NN1 of_IO the_AT theorem_NN1 ._. 
To_TO prove_VVI the_AT remainder_NN1 ,_, we_PPIS2 suppose_VV0 the_AT theorem_NN1 does_VDZ not_XX hold_VVI ;_; let_VV0 &lsqb;_( formula_NN1 &rsqb;_) have_VH0 p_NN1 positive_JJ and_CC r_ZZ1 p_ZZ1 negative_JJ elements_NN2 in_II31 addition_II32 to_II33 its_APPGE n_ZZ1 r_ZZ1 zeros_NN2 ,_, and_CC D_ZZ1 have_VH0 q_ZZ1 positive_JJ and_CC r_ZZ1 q_ZZ1 negative_JJ elements_NN2 ._. 
Then_RT &lsqb;_( formula_NN1 &rsqb;_) where_RRQ ,_, since_CS we_PPIS2 have_VH0 introduced_VVN the_AT negative_JJ signs_NN2 ,_, all_DB non-zero_JJ &lsqb;_( formula_NN1 &rsqb;_) i_MC1 and&lsqb;formula&rsqb;_NN1 are_VBR positive_JJ numbers_NN2 ._. 
We_PPIS2 begin_VV0 by_II supposing_VVG p_ZZ1 =_FO q_ZZ1 +_FO s_ZZ1 ,_, where_CS s_ZZ1 is_VBZ a_AT1 positive_JJ number_NN1 ,_, so_CS21 that_CS22 &lsqb;_( formula_NN1 &rsqb;_) ;_; and_CC we_PPIS2 examine_VV0 the_AT general_JJ quadratic_JJ form_NN1 &lsqb;_( formula_NN1 &rsqb;_) ._. 
If_CS &lsqb;_( formula_NN1 &rsqb;_) then_RT &lsqb;_( formula_NN1 &rsqb;_) These_DD2 two_MC expressions_NN2 are_VBR equal_JJ for_IF all_DB x_MC ;_; subtraction_NN1 yields_NN2 &lsqb;_( formula_NN1 &rsqb;_) Let_VV0 us_PPIO2 seek_VVI a_AT1 vector_NN1 x_ZZ1 such_CS21 that_CS22 &lsqb;_( formula_NN1 &rsqb;_) Now_RT from_II (_( 31_MC )_) ,_, &lsqb;_( formula_NN1 &rsqb;_) ;_; thus_RR the_AT first_MD part_NN1 of_IO (_( 33_MC )_) provides_VVZ n_ZZ1 p_ZZ1 relations_NN2 of_IO the_AT form_NN1 &lsqb;_( formula_NN1 &rsqb;_) ,_, and_CC the_AT second_MD part_NN1 ,_, q_ZZ1 similar_JJ relations_NN2 :_: a_AT1 total_NN1 of_IO n_ZZ1 p_ZZ1 +_FO q_ZZ1 =_FO n_ZZ1 s_ZZ1 relations_NN2 ._. 
But_CCB x_ZZ1 has_VHZ n_ZZ1 elements_NN2 ,_, so_CS21 that_CS22 we_PPIS2 can_VM find_VVI only_JJ n_ZZ1 s_ZZ1 of_IO its_APPGE elements_NN2 in_II31 terms_II32 of_II33 the_AT remaining_JJ s_ZZ1 (_( see_VV0 &lsqb;_( formula_NN1 &rsqb;_) an_AT1 example_NN1 is_VBZ given_VVN in_II 2.5_MC )_) which_DDQ are_VBR arbitrary_JJ ._. 
Thus_RR a_AT1 vector_NN1 x_ZZ1 satisfying_JJ (_( 33_MC )_) has_VHZ s_ZZ1 arbitrary_JJ elements_NN2 ._. 
Now_RT substitute_VV0 (_( 33_MC )_) into_II (_( 32_MC )_) ;_; the_AT RHS_NP1 vanishes_VVZ ,_, and_CC so_RR ,_, therefore_RR ,_, does_VDZ the_AT LHS_NP1 ._. 
With_IW all_RR &lsqb;_( formula_NN1 &rsqb;_) &lsqb;_( formula_NN1 &rsqb;_) positive_JJ this_DD1 implies_VVZ ,_, inter_RR21 alia_RR22 &lsqb;_( formula_NN1 &rsqb;_) This_DD1 ,_, taken_VVN together_RL with_IW (_( 33_MC )_) ,_, gives_VVZ y_ZZ1 =_FO 0_MC which_DDQ in_II31 view_II32 of_II33 (_( 31_MC )_) gives_VVZ x_ZZ1 =_FO 0_MC ._. 
We_PPIS2 thus_RR have_VH0 two_MC results_NN2 ;_; x_ZZ1 has_VHZ s_ZZ1 arbitrary_JJ elements_NN2 ,_, and_CC x_ZZ1 is_VBZ null_JJ :_: these_DD2 can_VM only_RR be_VBI reconciled_VVN if_CS s_ZZ1 =_FO 0_MC ,_, when_CS q_ZZ1 =_FO p_ZZ1 ._. 
If_CS we_PPIS2 now_RT take_VV0 &lsqb;_( formula_NN1 &rsqb;_) ,_, a_AT1 similar_JJ argument_NN1 leads_VVZ to_II the_AT same_DA result_NN1 q_ZZ1 =_FO p_ZZ1 ._. 
Since_CS the_AT argument_NN1 does_VDZ not_XX depend_VVI on_II the_AT nature_NN1 of_IO X_ZZ1 and_CC L_ZZ1 ,_, the_AT result_NN1 is_VBZ generally_RR true_JJ ,_, and_CC the_AT theorem_NN1 established_VVD ._. 
The_AT invariant_JJ p_ZZ1 is_VBZ known_VVN as_II the_AT index_NN1 of_IO positiveness_NN1 ,_, or_CC simply_RR the_AT index_NN1 ,_, of_IO A_ZZ1 and_CC its_APPGE quadratic_JJ form_NN1 ._. 
Examples_NN2 XV_MC (_( a_ZZ1 )_) Let_VV0 &lsqb;_( formula_NN1 &rsqb;_) then_RT its_APPGE orthogonal_JJ matrix_NN1 X_ZZ1 is_VBZ &lsqb;_( formula_NN1 &rsqb;_) which_DDQ yields_VVZ &lsqb;_( formula_NN1 &rsqb;_) We_PPIS2 build_VV0 up_RP L_ZZ1 as_II the_AT product_NN1 &lsqb;_( formula_NN1 &rsqb;_) ,_, where_CS L1_FO annihilates_VVZ the_AT last_MD three_MC elements_NN2 of_IO the_AT first_MD column_NN1 of_IO A_ZZ1 ,_, L2_FO the_AT last_MD two_MC elements_NN2 of_IO the_AT second_MD column_NN1 ,_, and_CC L3_FO the_AT last_MD element_NN1 of_IO the_AT third_MD column_NN1 ;_; then_RT &lsqb;_( formula_NN1 &rsqb;_) Then_RT &lsqb;_( formula_NN1 &rsqb;_) and_CC the_AT diagonal_JJ elements_NN2 of_IO LA_NP1 are_VBR those_DD2 of_IO D_ZZ1 =_FO &lsqb;_( formula_NN1 &rsqb;_) We_PPIS2 see_VV0 that_CST both_RR &lsqb;_( formula_NN1 &rsqb;_) and_CC D_ZZ1 have_VH0 two_MC positive_JJ and_CC two_MC negative_JJ elements_NN2 ._. 
It_PPH1 may_VM be_VBI noted_VVN that_CST &lsqb;_( formula_NN1 &rsqb;_) ._. 
(_( b_ZZ1 )_) Let_VV0 us_PPIO2 add_VVI 2_MC to_II each_DD1 diagonal_JJ element_NN1 of_IO A_ZZ1 ,_, to_TO give_VVI &lsqb;_( formula_NN1 &rsqb;_) ._. 
then_RT we_PPIS2 know_VV0 (_( Theorem_NN1 IV_MC )_) that_CST &lsqb;_( formula_NN1 &rsqb;_) ._. 
We_PPIS2 leave_VV0 it_PPH1 to_II the_AT reader_NN1 to_TO construct_VVI the_AT corresponding_JJ Lo_FW ;_; and_CC to_TO show_VVI that_CST &lsqb;_( formula_NN1 &rsqb;_) Thus_RR both_RR &lsqb;_( formula_NN1 &rsqb;_) and_CC &lsqb;_( formula_NN1 &rsqb;_) have_VH0 two_MC positive_JJ ,_, one_MC1 zero_NN1 ,_, and_CC one_MC1 negative_JJ element_NN1 ._. 
Since_CS Ao_NP1 is_VBZ singular_JJ ,_, so_RR is_VBZ &lsqb;_( formula_NN1 &rsqb;_) which_DDQ is_VBZ shown_VVN by_II the_AT null_JJ row._NNU 2_MC SOME_DD NUMERICAL_JJ METHODS_NN2 2.1_MC INTRODUCTION_NN1 In_II this_DD1 chapter_NN1 we_PPIS2 shall_VM discuss_VVI some_DD numerical_JJ methods_NN2 for_IF solving_VVG problems_NN2 formulated_VVN in_II31 terms_II32 of_II33 matrices_NN2 ._. 
However_RR ,_, it_PPH1 must_VM be_VBI stressed_VVN at_II the_AT outset_NN1 that_CST we_PPIS2 can_VM do_VDI no_AT more_RRR here_RL than_CSN to_TO indicate_VVI the_AT basic_JJ principles_NN2 involved_VVD ,_, and_CC give_VV0 illustrations_NN2 of_IO a_AT1 few_DA2 of_IO the_AT almost_RR innumerable_JJ variants_NN2 of_IO basic_JJ methods_NN2 that_CST exist_VV0 ._. 
Quot_VV0 homines_NN2 ,_, tot_NN1 sententiae_NN2 :_: almost_RR every_AT1 worker_NN1 in_II the_AT field_NN1 has_VHZ his_APPGE pet_NN1 variation_NN1 ,_, which_DDQ suits_VVZ him_PPHO1 ad_NN1 his_APPGE computer_NN1 better_RRR than_CSN any_DD other_JJ ._. 
For_IF a_AT1 wider_JJR discussion_NN1 ,_, readers_NN2 are_VBR referred_VVN to_II texts_NN2 such_II21 as_II22 Bodewig_NP1 (_( 5_MC )_) and_CC Wilkinson_NP1 (_( 6_MC )_) ._. 
Most_DAT of_IO the_AT numerical_JJ problems_NN2 which_DDQ arise_VV0 in_II matrix_NN1 studies_NN2 (_( particularly_RR those_DD2 which_DDQ derive_VV0 from_II dynamical_JJ systems_NN2 )_) lead_VV0 ultimately_RR to_II one_MC1 or_CC both_DB2 of_IO two_MC basic_JJ processes_NN2 :_: (_( a_ZZ1 )_) the_AT inversion_NN1 of_IO a_AT1 numerical_JJ matrix_NN1 ;_; (_( b_ZZ1 )_) the_AT evaluation_NN1 of_IO its_APPGE eigenvalues_NN2 and_CC vectors_NN2 ._. 
It_PPH1 will_VM be_VBI convenient_JJ to_TO deal_VVI with_IW these_DD2 under_II separate_JJ headings_NN2 ._. 
The_AT methods_NN2 themselves_PPX2 also_RR divide_VV0 into_II two_MC broad_JJ categories_NN2 :_: (_( a_ZZ1 )_) direct_JJ methods_NN2 ,_, in_II which_DDQ the_AT solution_NN1 requires_VVZ is_VBZ reached_VVN in_II a_AT1 finite_JJ ,_, predictable_JJ number_NN1 of_IO operations_NN2 ;_; (_( b_ZZ1 )_) indirect_JJ methods_NN2 ,_, in_II which_DDQ the_AT solution_NN1 is_VBZ usually_RR obtained_VVN by_II successive_JJ approximation_NN1 procedures_NN2 ,_, when_CS the_AT number_NN1 of_IO operations_NN2 is_VBZ determined_VVN by_II the_AT rate_NN1 of_IO convergence_NN1 ._. 
I_MC1 :_: RECIPROCATION_NP1 ;_; LINEAR_JJ ALGEBRAIC_JJ EQUATIONS_NN2 2.2_MC DIRECT_JJ METHOD_NN1 FOR_IF MATRIX_NN1 INVERSION_NN1 Under_II this_DD1 heading_NN1 we_PPIS2 give_VV0 three_MC methods_NN2 which_DDQ are_VBR all_DB very_RG different_JJ in_II principle_NN1 ._. 
Each_DD1 has_VHZ variants_NN2 ;_; for_IF each_DD1 there_EX are_VBR circumstances_NN2 which_DDQ favour_VV0 its_APPGE use._NNU 2.2.1_MC ._. 
Pivotal_JJ condensation_NN1 This_DD1 method_NN1 is_VBZ the_AT simplest_JJT ,_, and_CC probably_RR the_AT most_RGT universally_RR used_VVN ._. 
Suppose_VV0 we_PPIS2 have_VH0 a_AT1 square_JJ matrix_NN1 A_ZZ1 =_FO &lsqb;_( formula_NN1 &rsqb;_) of_IO order_NN1 n_ZZ1 for_IF which_DDQ we_PPIS2 require_VV0 to_TO find_VVI the_AT reciprocal_JJ R_ZZ1 :_: then_RT AR_UH =_FO I._NP1 Now_RT it_PPH1 is_VBZ possible_JJ to_TO write_VVI down_RP a_AT1 sequence_NN1 of_IO matrices_NN2 &lsqb;_( formula_NN1 &rsqb;_) Mr_NNB (_( r_ZZ1 is_VBZ usually_RR n_ZZ1 or_CC n_ZZ1 +_FO 1_MC1 ,_, depending_II21 on_II22 the_AT variant_NN1 employed_VVN )_) which_DDQ when_CS used_VVN as_II a_AT1 chain_NN1 to_II premultiply_RR A_ZZ1 ,_, condense_VV0 it_PPH1 to_II the_AT unit_NN1 matrix_NN1 ,_, so_CS21 that_CS22 &lsqb;_( formula_NN1 &rsqb;_) which_DDQ evidently_RR implies_VVZ &lsqb;_( formula_NN1 &rsqb;_) The_AT numerical_JJ procedure_NN1 is_VBZ to_TO operate_VVI successively_RR on_II 91_MC )_) ;_; given_VVN A_ZZ1 we_PPIS2 can_VM write_VVI down_RP M1_FO and_CC we_PPIS2 evaluate_VV0 &lsqb;_( formula_NN1 &rsqb;_) so_CS21 that_CS22 (_( 1_MC1 )_) becomes_VVZ &lsqb;_( formula_NN1 &rsqb;_) Knowing_VVG B_ZZ1 we_PPIS2 can_VM now_RT write_VVI down_RP &lsqb;_( formula_NN1 &rsqb;_) and_CC form_NN1 &lsqb;_( formula_NN1 &rsqb;_) and_RR31 so_RR32 on_RR33 ,_, until_CS R_ZZ1 is_VBZ obtained_VVN ._. 
As_CSA we_PPIS2 shall_VM see_VVI ,_, in_II practice_NN1 it_PPH1 is_VBZ not_XX necessary_JJ to_TO write_VVI down_RP the_AT matrices_NN2 &lsqb;_( formula_NN1 &rsqb;_) merely_RR perform_VV0 simple_JJ operations_NN2 with_IW the_AT rows_NN2 of_IO A_ZZ1 and_CC I_ZZ1 in_II (_( 1_MC1 )_) ,_, which_DDQ is_VBZ all_DB the_AT matrices_NN2 &lsqb;_( formula_NN1 &rsqb;_) do_VD0 ._. 
For_IF ease_NN1 of_IO exposition_NN1 ,_, let_VV0 us_PPIO2 suppose_VVI n_ZZ1 =_FO 4_MC ._. 
Consider_VV0 the_AT matrix_NN1 product_NN1 ,_, corresponding_VVG to_II (_( 3_MC )_) &lsqb;_( formula_NN1 &rsqb;_) In_II writing_VVG down_RP &lsqb;_( formula_NN1 &rsqb;_) we_PPIS2 have_VH0 first_MD arbitrarily_RR selected_VVN a_AT1 non-zero_JJ element_NN1 &lsqb;_( formula_NN1 &rsqb;_) of_IO A_ZZ1 as_II a_AT1 pivot_NN1 (_( underlined_VVN )_) for_IF this_DD1 first_MD step_NN1 in_II the_AT operation._NNU &lsqb;_( formula_NN1 &rsqb;_) is_VBZ then_RT obtained_VVN from_II the_AT unit_NN1 matrix_NN1 by_II replacing_VVG its_APPGE second_MD column_NN1 (_( corresponding_VVG to_II the_AT subscript_NN1 2_MC of_IO &lsqb;_( formula_NN1 &rsqb;_) by_II the_AT third_MD column_NN1 of_IO A_ZZ1 divided_VVN through_RP by_II &lsqb;_( formula_NN1 &rsqb;_) with_IW negative_JJ signs_NN2 for_IF the_AT off-diagonal_JJ elements_NN2 ._. 
The_AT third_MD column_NN1 of_IO B_ZZ1 is_VBZ then_RT null_JJ except_II21 for_II22 a_AT1 unit_NN1 in_II the_AT position_NN1 corresponding_VVG to_II &lsqb;_( formula_NN1 &rsqb;_) The_AT product_NN1 makes_VVZ it_PPH1 clear_JJ that_CST all_DB we_PPIS2 are_VBR doing_VDG ,_, in_II fact_NN1 ,_, is_VBZ to_TO operate_VVI with_IW the_AT rows_NN2 of_IO A_ZZ1 in_II a_AT1 manner_NN1 similar_JJ to_II that_DD1 often_RR used_VVN in_II the_AT condensation_NN1 of_IO determinants_NN2 ._. 
Thus_RR row_VV0 1_MC1 of_IO B_ZZ1 is_VBZ now_RT 1_MC1 of_IO A_AT1 minus_NN1 &lsqb;_( formula_NN1 &rsqb;_) times_II row_NN1 2_MC of_IO A_ZZ1 ;_; row_VV0 2_MC of_IO B_ZZ1 is_VBZ row_NN1 2_MC of_IO A_ZZ1 divided_VVN throughout_RL by_II &lsqb;_( formula_NN1 &rsqb;_) 3_MC ;_; row_VV0 3_MC of_IO B_ZZ1 is_VBZ row_NN1 3_MC of_IO A_AT1 minus_NN1 &lsqb;_( formula_NN1 &rsqb;_) times_II row_NN1 2_MC of_IO A_ZZ1 ;_; and_RR31 so_RR32 on_RR33 ._. 
Having_VHG obtained_VVN B_ZZ1 ,_, we_PPIS2 now_RT select_VV0 a_AT1 pivotal_JJ element_NN1 in_II it_PPH1 for_IF the_AT next_MD step_NN1 ;_; the_AT selection_NN1 is_VBZ arbitrary_JJ except_CS21 that_CS22 the_AT element_NN1 must_VM be_VBI nonzero_NN1 and_CC must_VM not_XX be_VBI in_II the_AT second_MD row_NN1 ,_, which_DDQ contains_VVZ the_AT unit_NN1 ._. 
For_REX21 example_REX22 &lsqb;_( formula_NN1 &rsqb;_) in_II which_DDQ &lsqb;_( formula_NN1 &rsqb;_) has_VHZ been_VBN chosen_VVN as_II the_AT pivotal_JJ element_NN1 ._. 
We_PPIS2 now_RT choose_VV0 a_AT1 pivotal_JJ element_NN1 in_II C_NP1 ,_, excluding_VVG rows_NN2 2_MC and_CC 3_MC containing_VVG the_AT units_NN2 ._. 
Note_VV0 that_CST &lsqb;_( formula_NN1 &rsqb;_) and_CC &lsqb;_( formula_NN1 &rsqb;_) can_VM not_XX both_RR be_VBI zero_MC ,_, or_CC A_ZZ1 would_VM be_VBI singular_JJ and_CC R_ZZ1 would_VM not_XX exist_VVI ;_; similarly_RR for_IF &lsqb;_( formula_NN1 &rsqb;_) We_PPIS2 choose_VV0 a_AT1 nonzero_NN1 element_NN1 from_II these_DD2 four_MC -say_JJ &lsqb;_( formula_NN1 &rsqb;_) and_CC then_RT &lsqb;_( formula_NN1 &rsqb;_) The_AT last_MD step_NN1 is_VBZ not_XX arbitrary_JJ :_: we_PPIS2 must_VM exclude_VVI the_AT first_MD three_MC rows_NN2 (_( containing_VVG units_NN2 )_) ,_, so_CS21 that_CS22 the_AT only_JJ choice_NN1 for_IF pivot_NN1 in_II the_AT next_MD step_NN1 is_VBZ D44_FO ,_, which_DDQ can_VM not_XX be_VBI zero_MC ._. 
We_PPIS2 write_VV0 down_RP &lsqb;_( formula_NN1 &rsqb;_) ,_, evaluate_VV0 E_ZZ1 =_FO &lsqb;_( formula_NN1 &rsqb;_) obtaining_VVG a_AT1 matrix_NN1 which_DDQ if_CS the_AT rows_NN2 are_VBR rearranged_VVN by_II premultiplication_NN1 by_II an_AT1 obvious_JJ matrix_NN1 &lsqb;_( formula_NN1 &rsqb;_) ,_, becomes_VVZ the_AT unit_NN1 matrix_NN1 ._. 
As_CSA we_PPIS2 have_VH0 said_VVN ,_, in_II practice_NN1 we_PPIS2 do_VD0 not_XX need_VVI to_TO write_VVI down_RP M1_FO etc_RA ._. 
When_CS we_PPIS2 use_VV0 &lsqb;_( formula_NN1 &rsqb;_) as_II a_AT1 premultiplier_JJR of_IO (_( 1_MC1 )_) we_PPIS2 combine_VV0 the_AT rows_NN2 of_IO I_ZZ1 in_II exactly_RR the_AT same_DA ways_NN2 as_CSA those_DD2 of_IO A._NP1 We_PPIS2 can_VM therefore_RR write_VVI the_AT two_MC arrays_NN2 side_NN1 by_II side_NN1 and_CC operate_VV0 on_II rows_NN2 of_IO 2n_FO elements_NN2 ._. 
For_REX21 example_REX22 ,_, to_TO summarise_VVI what_DDQ has_VHZ been_VBN done_VDN above_RL (_( we_PPIS2 refer_VV0 to_TO row_VVI i_MC1 as_CSA &lsqb;_( formula_NN1 &rsqb;_) )_) we_PPIS2 use_VV0 Table_NN1 1._MC &lsqb;_( formula_NN1 &rsqb;_) When_RRQ the_AT left-hand_JJ array_NN1 has_VHZ been_VBN condensed_VVN to_II I_ZZ1 ,_, the_AT right_JJ hand_NN1 array_NN1 is_VBZ R._NP1 This_DD1 method_NN1 has_VHZ a_AT1 number_NN1 of_IO variants_NN2 ._. 
First_MD ,_, it_PPH1 is_VBZ possible_JJ to_TO operate_VVI with_IW columns_NN2 ,_, using_VVG postmultiplying_JJ matrices_NN2 Mi_NP1 ;_; but_CCB this_DD1 is_VBZ really_RR only_JJ transposition_NN1 of_IO operations_NN2 on_II rows_NN2 ._. 
The_AT variants_NN2 derive_VV0 mainly_RR from_II the_AT choice_NN1 of_IO pivot_NN1 ._. 
In_II the_AT so-called_JJ optimal_JJ pivoting_JJ method_NN1 ,_, one_PN1 chooses_VVZ the_AT element_NN1 of_IO largest_JJT modulus_NN1 from_II among_II those_DD2 available_JJ ._. 
In_II a_AT1 second_MD variant_NN1 ,_, one_PN1 chooses_VVZ the_AT element_NN1 of_IO largest_JJT modulus_NN1 in_II each_DD1 column_NN1 in_II turn_NN1 ;_; in_II a_AT1 third_MD ,_, the_AT diagonal_JJ elements_NN2 are_VBR chosen_VVN in_II order_NN1 ;_; and_CC in_II a_AT1 fourth_MD ,_, the_AT diagonal_JJ elements_NN2 of_IO largest_JJT modulus_NN1 are_VBR chosen_VVN in_II turn_NN1 ._. 
The_AT third_MD has_VHZ the_AT advantage_NN1 that_CST no_AT final_JJ rearrangement_NN1 of_IO rows_NN2 is_VBZ required_VVN ;_; on_II the_AT other_JJ hand_NN1 ,_, it_PPH1 fails_VVZ if_CS a_AT1 pivot_NN1 becomes_VVZ zero_MC ,_, or_CC even_RR very_RG small_JJ ._. 
The_AT third_MD variant_NN1 is_VBZ usually_RR described_VVN as_II &quot;_" diagonal_JJ pivoting_JJ without_IW row_NN1 interchange_NN1 &quot;_" ,_, the_AT fourth_MD as_CSA &quot;_" diagonal_JJ pivoting_JJ with_IW pivotal_JJ strategy_NN1 &quot;_" ._. 
It_PPH1 is_VBZ to_TO be_VBI observed_VVN that_CST the_AT determinant_NN1 of_IO A_ZZ1 is_VBZ readily_RR found_VVN from_II the_AT condensation_NN1 procedure_NN1 ._. 
The_AT determinant_NN1 of_IO &lsqb;_( formula_NN1 &rsqb;_) is_VBZ &lsqb;_( formula_NN1 &rsqb;_) ,_, etc_RA ._. 
It_PPH1 follows_VVZ that_CST the_AT determinant_NN1 of_IO A_ZZ1 is_VBZ &lsqb;_( formula_NN1 &rsqb;_) ,_, i.e._REX the_AT product_NN1 of_IO the_AT pivotal_JJ elements_NN2 chosen_VVN ,_, multiplied_VVN by_II the_AT determinant_NN1 of_IO &lsqb;_( formula_NN1 &rsqb;_) ,_, which_DDQ rearranges_VVZ the_AT rows_NN2 ,_, and_CC so_RR has_VHZ the_AT value_NN1 &lsqb;_( formula_NN1 &rsqb;_) ,_, according_CS21 as_CS22 to_II whether_CSW the_AT number_NN1 of_IO interchanges_NN2 is_VBZ even_RR or_CC odd_JJ ._. 
Example_NN1 1_MC1 We_PPIS2 exemplify_VV0 these_DD2 procedures_NN2 using_VVG the_AT matrix_NN1 &lsqb;_( formula_NN1 &rsqb;_) This_DD1 matrix_NN1 ,_, in_II fact_NN1 ,_, has_VHZ the_AT determinant_NN1 100_MC ;_; since_CS its_APPGE elements_NN2 are_VBR integers_NN2 ,_, the_AT elements_NN2 of_IO the_AT reciprocal_JJ will_VM be_VBI terminating_JJ decimals_NN2 ._. 
In_II the_AT intermediate_JJ steps_NN2 ,_, however_RR ,_, the_AT numbers_NN2 are_VBR incommensurable_JJ :_: for_IF brevity_NN1 of_IO exposition_NN1 we_PPIS2 give_VV0 four_MC decimal_JJ places_NN2 only_RR ,_, though_CS of_RR21 course_RR22 a_AT1 computer_NN1 will_VM normally_RR employ_VVI a_AT1 much_RR greater_JJR number._NNU (_( i_ZZ1 )_) Optimal_JJ pivoting_JJ In_II Table_NN1 2_MC ,_, the_AT last_MD array_NN1 on_II the_AT right_NN1 is_VBZ the_AT required_JJ reciprocal_JJ ._. 
Note_VV0 that_CST ,_, in_II the_AT rearrangement_NN1 of_IO the_AT last_MD step_NN1 ,_, we_PPIS2 crossed_VVD three_MC rows_NN2 in_II bringing_VVG &lsqb;_( formula_NN1 &rsqb;_) to_II the_AT top_NN1 ;_; then_RT the_AT new_JJ second_MD row_NN1 &lsqb;_( formula_NN1 &rsqb;_) crossed_VVD two_MC rows_NN2 in_II going_VVG to_II the_AT bottom_NN1 ._. 
Since_CS the_AT total_JJ number_NN1 of_IO crossings_NN2 ,_, 5_MC is_VBZ odd_JJ ,_, &lsqb;_( formula_NN1 &rsqb;_) has_VHZ the_AT determinant_NN1 -1_MC ;_; accordingly_RR &lsqb;_( formula_NN1 &rsqb;_) &lsqb;_( formula_NN1 &rsqb;_) &lsqb;_( formula_NN1 &rsqb;_) Coincidentally_RR ,_, this_DD1 example_NN1 also_RR illustrates_VVZ he_PPHS1 second_MD variant_NN1 :_: it_PPH1 is_VBZ adventitious_JJ that_CST the_AT largest_JJT element_NN1 among_II those_DD2 available_JJ occurs_VVZ in_II the_AT first_MD ,_, second_NNT1 ,_, third_MD and_CC fourth_MD column_NN1 in_II succession_NN1 ._. 
There_EX is_VBZ thus_RR no_AT difference_NN1 in_II the_AT working_JJ ._. 
However_RR ,_, in_II the_AT optimum_JJ method_NN1 a_AT1 computer_NN1 is_VBZ required_VVN to_TO scan_VVI all_DB elements_NN2 available_JJ ,_, i.e._REX &lsqb;_( formula_NN1 &rsqb;_) in_II successive_JJ steps_NN2 ,_, while_CS in_II the_AT second_MD variant_NN1 only_RR m_ZZ1 ,_, (_( n_ZZ1 1_MC1 )_) ,_, ..._... ,_, 1_MC1 need_VV0 scanning_VVG ._. 
In_II this_DD1 example_NN1 ,_, therefore_RR ,_, the_AT second_MD variant_NN1 has_VHZ the_AT advantage_NN1 ;_; in_II a_AT1 matrix_NN1 of_IO large_JJ order_NN1 ,_, the_AT scanning_NN1 process_NN1 can_VM take_VVI considerable_JJ computer_NN1 time_NNT1 ._. 
No_AT scanning_NN1 is_VBZ needed_VVN in_II the_AT third_MD variant_NN1 ,_, where_CS the_AT diagonal_JJ elements_NN2 are_VBR prescribed._NNU (_( ii_MC )_) Diagonal_JJ pivoting_JJ We_PPIS2 now_RT treat_VV0 the_AT same_DA matrix_NN1 by_II the_AT third_MD variant_NN1 (_( see_VV0 Table_NN1 3_MC )_) ._. 
For_IF this_DD1 example_NN1 ,_, this_DD1 variant_NN1 presents_VVZ no_AT difficulty_NN1 ._. 
Also_RR &lsqb;_( formula_NN1 &rsqb;_) as_CSA before_RT ._. 
The_AT student_NN1 is_VBZ invited_VVN to_TO effect_VVI this_DD1 reciprocation_NN1 by_II31 means_II32 of_II33 the_AT fourth_MD variant_NN1 ,_, in_II which_DDQ the_AT largest_JJT diagonal_JJ element_NN1 available_JJ is_VBZ chosen_VVN as_II the_AT pivot_NN1 at_II each_DD1 stage_NN1 ;_; it_PPH1 begins_VVZ with_IW the_AT bottom_JJ right-hand_JJ element._NNU 2.2.2_MC The_AT method_NN1 of_IO submatrices_NN2 This_DD1 method_NN1 has_VHZ nothing_PN1 in_II31 common_II32 with_II33 the_AT pivotal_JJ method_NN1 ,_, but_CCB has_VHZ great_JJ advantages_NN2 in_II certain_JJ circumstances._NNU (_( a_ZZ1 )_) Suppose_VV0 we_PPIS2 have_VH0 a_AT1 computer_NN1 which_DDQ will_VM accommodate_VVI a_AT1 matrix_NN1 of_IO order_NN1 m_ZZ1 ,_, but_CCB we_PPIS2 require_VV0 to_TO reciprocate_VVI a_AT1 larger_JJR matrix_NN1 A_ZZ1 of_IO order_NN1 n_ZZ1 where_RRQ &lsqb;_( formula_NN1 &rsqb;_) ._. 
(_( b_ZZ1 )_) Suppose_VV0 we_PPIS2 have_VH0 a_AT1 matrix_NN1 B_ZZ1 of_IO which_DDQ we_PPIS2 know_VV0 the_AT reciprocal_JJ &lsqb;_( formula_NN1 &rsqb;_) ;_; B_ZZ1 is_VBZ now_RT bordered_VVN ,_, so_CS21 that_CS22 it_PPH1 becomes_VVZ a_AT1 submatrix_NN1 of_IO a_AT1 larger_JJR matrix_NN1 A_ZZ1 ,_, and_CC we_PPIS2 require_VV0 A-1_MC1 ._. 
The_AT method_NN1 of_IO submatrices_NN2 is_VBZ particularly_RR useful_JJ in_II solving_VVG both_DB2 these_DD2 problems_NN2 ._. 
Let_VV0 A_ZZ1 be_VBI partitioned_VVN in_II the_AT form_NN1 &lsqb;_( formula_NN1 &rsqb;_) Here_RL A_ZZ1 is_VBZ of_IO order_NN1 n_ZZ1 ,_, B_ZZ1 of_IO order_NN1 m_ZZ1 ,_, E_ZZ1 of_IO order_NN1 n_ZZ1 m_ZZ1 ;_; C_NP1 and_CC D_ZZ1 are_VBR in_RR21 general_RR22 rectangular_JJ ._. 
Let_VV0 the_AT reciprocal_JJ of_IO A_ZZ1 be_VBI similarly_RR partitioned_VVN :_: &lsqb;_( formula_NN1 &rsqb;_) Then_RT the_AT equations_NN2 &lsqb;_( formula_NN1 &rsqb;_) provide_VV0 relations_NN2 from_II which_DDQ ,_, in_RR21 general_RR22 ,_, &lsqb;_( formula_NN1 &rsqb;_) &lsqb;_( formula_NN1 &rsqb;_) ,_, may_VM be_VBI explicitly_RR found_VVN ._. 
Various_JJ formulae_NN2 are_VBR possible_JJ ,_, but_CCB the_AT most_RGT advantageous_JJ from_II the_AT computational_JJ standpoint_NN1 appear_VV0 to_TO be_VBI the_AT following_JJ ._. 
Write_VV0 &lsqb;_( formula_NN1 &rsqb;_) Then_RT &lsqb;_( formula_NN1 &rsqb;_) as_CSA may_VM be_VBI checked_VVN by_II evaluating_VVG &lsqb;_( formula_NN1 &rsqb;_) It_PPH1 is_VBZ to_TO be_VBI noted_VVN that_CST ,_, although_CS these_DD2 formulae_NN2 involve_VV0 a_AT1 number_NN1 of_IO multiplications_NN2 ,_, only_RR two_MC reciprocations_NN2 ,_, B-1_FO (_( of_IO order_NN1 m_ZZ1 )_) and_CC &lsqb;_( formula_NN1 &rsqb;_) (_( of_IO order_NN1 n_ZZ1 m_ZZ1 )_) are_VBR involved_VVN ._. 
Problem_NN1 (_( a_ZZ1 )_) It_PPH1 is_VBZ clear_JJ that_CST the_AT formulae_NN2 just_RR established_VVN provide_VV0 a_AT1 method_NN1 of_IO solution_NN1 for_IF problem_NN1 (_( a_ZZ1 )_) since_CS &lsqb;_( formula_NN1 &rsqb;_) and_CC the_AT computer_NN1 can_VM reciprocate_VVI both_RR B_ZZ1 and_CC &lsqb;_( formula_NN1 &rsqb;_) and_CC perform_VV0 the_AT required_JJ multiplications_NN2 ._. 
Example_NN1 1_MC1 Although_CS it_PPH1 is_VBZ very_RG small_JJ ,_, we_PPIS2 may_VM use_VVI the_AT matrix_NN1 A_ZZ1 of_IO Equation_NN1 (_( 2.2.1.4_MC )_) to_TO illustrate_VVI the_AT principles_NN2 involved_VVD ._. 
We_PPIS2 shall_VM here_RL partition_VVI it_PPH1 into_II four_MC square_JJ submatrices_NN2 ,_, each_DD1 of_IO order_NN1 2_MC ._. 
In_II this_DD1 simple_JJ illustrative_JJ case_NN1 we_PPIS2 avoid_VV0 decimals_NN2 ._. 
Thus_RR &lsqb;_( formula_NN1 &rsqb;_) &lsqb;_( formula_NN1 &rsqb;_) We_PPIS2 are_VBR now_RT able_JK to_TO evaluate_VVI the_AT remaining_JJ submatrices_NN2 of_IO the_AT reciprocal_JJ as_CSA &lsqb;_( formula_NN1 &rsqb;_) Accordingly_RR ,_, the_AT reciprocal_JJ is_VBZ &lsqb;_( formula_NN1 &rsqb;_) in_II agreement_NN1 with_IW the_AT result_NN1 found_VVD earlier_RRR ._. 
Problem_NN1 (_( b_ZZ1 )_) It_PPH1 is_VBZ also_RR clear_VV0 that_CST Equations_NN2 (_( 3_MC )_) and_CC (_( 4_MC )_) provide_VV0 a_AT1 solution_NN1 for_IF this_DD1 problem_NN1 ._. 
We_PPIS2 are_VBR given_VVN &lsqb;_( formula_NN1 &rsqb;_) and_CC we_PPIS2 border_VV0 B_ZZ1 as_CSA in_II (_( 1_MC1 )_) and_CC apply_VV0 our_APPGE solution_NN1 to_TO obtain_VVI &lsqb;_( formula_NN1 &rsqb;_) The_AT working_NN1 is_VBZ then_RT exactly_RR as_CSA in_II the_AT example_NN1 just_RR given_VVN ._. 
The_AT formulae_NN2 are_VBR ,_, however_RR ,_, rendered_VVN much_RR simpler_JJR if_CS we_PPIS2 have_VH0 only_JJ line_NN1 bordering_VVG ,_, i.e._REX C_ZZ1 is_VBZ a_AT1 column_NN1 ,_, D_ZZ1 a_AT1 row_NN1 ,_, and_CC E_ZZ1 a_AT1 scalar_JJ ._. 
Then_RT X_ZZ1 is_VBZ a_AT1 column_NN1 ,_, Y_ZZ1 a_AT1 raw_JJ ,_, and_CC &lsqb;_( formula_NN1 &rsqb;_) a_AT1 scalar_JJ ._. 
It_PPH1 follows_VVZ that_CST we_PPIS2 can_VM provide_VVI the_AT answer_NN1 to_II a_AT1 third_MD problem_NN1 ,_, (_( c_ZZ1 )_) :_: the_AT successive_JJ evaluation_NN1 of_IO the_AT reciprocals_NN2 of_IO the_AT leading_JJ minors_NN2 of_IO a_AT1 matrix_NN1 A_ZZ1 until_CS &lsqb;_( formula_NN1 &rsqb;_) is_VBZ achieved_VVN ._. 
Example_NN1 2_MC Again_RT ,_, we_PPIS2 choose_VV0 the_AT matrix_NN1 A_ZZ1 of_IO Equation_NN1 (_( 2.2.1.4_MC )_) :_: (_( i_ZZ1 )_) The_AT leading_JJ minor_NN1 of_IO order_NN1 1_MC1 ,_, b1_FO ,_, is_VBZ the_AT element_NN1 6_MC :_: reciprocal_JJ 1/6._MF (_( ii_MC )_) Using_VVG (_( i_ZZ1 )_) ,_, we_PPIS2 apply_VV0 the_AT formulae_NN2 (_( 3_MC )_) ,_, (_( 4_MC )_) to_TO find_VVI the_AT reciprocal_JJ of_IO the_AT leading_JJ minor_NN1 of_IO order_NN1 2_MC ,_, viz_REX :_: &lsqb;_( formula_NN1 &rsqb;_) Here_RL &lsqb;_( formula_NN1 &rsqb;_) and_CC we_PPIS2 deduce_VV0 &lsqb;_( formula_NN1 &rsqb;_) (_( iii_MC )_) We_PPIS2 now_RT use_VV0 this_DD1 result_NN1 to_TO examine_VVI the_AT third_MD order_NN1 minor_NN1 &lsqb;_( formula_NN1 &rsqb;_) Then_RT &lsqb;_( formula_NN1 &rsqb;_) similarly_RR ,_, &lsqb;_( formula_NN1 &rsqb;_) Also_RR &lsqb;_( formula_NN1 &rsqb;_) Note_VV0 that_CST ,_, in_II Equations_NN2 (_( 4_MC )_) which_DDQ we_PPIS2 now_RT evaluate_VV0 ,_, &lsqb;_( formula_NN1 &rsqb;_) is_VBZ a_AT1 matrix_NN1 of_IO rank_NN1 1_MC1 ._. 
We_PPIS2 find_VV0 &lsqb;_( formula_NN1 &rsqb;_) (_( iv_MC )_) Using_VVG &lsqb;_( formula_NN1 &rsqb;_) ,_, we_PPIS2 proceed_VV0 to_TO obtain_VVI the_AT reciprocal_JJ of_IO &lsqb;_( formula_NN1 &rsqb;_) ._. 
We_PPIS2 leave_VV0 this_DD1 step_NN1 to_II the_AT reader_NN1 ,_, but_CCB observe_VV0 that_CST &lsqb;_( formula_NN1 &rsqb;_) Since_CS an_AT1 element_NN1 in_II the_AT reciprocal_JJ of_IO a_AT1 matrix_NN1 M_ZZ1 is_VBZ defined_VVN as_II the_AT ratio_NN1 of_IO the_AT cofactor_NN1 of_IO the_AT corresponding_JJ element_NN1 in_II M_NN1 to_II the_AT determinant_NN1 &lsqb;_( formula_NN1 &rsqb;_) ,_, it_PPH1 follows_VVZ that_CST ,_, for_REX21 example_REX22 ,_, &lsqb;_( formula_NN1 &rsqb;_) and_CC hence_RR &lsqb;_( formula_NN1 &rsqb;_) In_II this_DD1 example_NN1 ,_, therefore_RR ,_, &lsqb;_( formula_NN1 &rsqb;_) 2.2.3_MC Column_NN1 (_( or_CC row_NN1 )_) building_VVG This_DD1 method_NN1 makes_VVZ use_NN1 of_IO Theorem_NN1 IX_MC of_IO &lsqb;_( formula_NN1 &rsqb;_) which_DDQ states_VVZ that_CST if_CS we_PPIS2 have_VH0 a_AT1 square_JJ matrix_NN1 Ao_NP1 of_IO order_NN1 n_ZZ1 ,_, and_CC know_VV0 &lsqb;_( formula_NN1 &rsqb;_) ,_, then_RT by_II a_AT1 simple_JJ calculation_NN1 we_PPIS2 can_VM find_VVI the_AT reciprocal_JJ &lsqb;_( formula_NN1 &rsqb;_) of_IO a_AT1 new_JJ matrix_NN1 &lsqb;_( formula_NN1 &rsqb;_) ,_, where_RRQ &lsqb;_( formula_NN1 &rsqb;_) is_VBZ a_AT1 unit_NN1 rank_NN1 matrix_NN1 ,_, also_RR of_IO order_NN1 n_ZZ1 ,_, &lsqb;_( formula_NN1 &rsqb;_) being_VBG an_AT1 arbitrary_JJ column_NN1 and_CC row_VV0 respectively_RR ._. 
A_AT1 second_MD application_NN1 of_IO the_AT theorem_NN1 enables_VVZ us_PPIO2 to_TO find_VVI &lsqb;_( formula_NN1 &rsqb;_) ,_, where_RRQ &lsqb;_( formula_NN1 &rsqb;_) and_RR31 so_RR32 on_RR33 ,_, until_CS we_PPIS2 find_VV0 &lsqb;_( formula_NN1 &rsqb;_) ,_, where_RRQ &lsqb;_( formula_NN1 &rsqb;_) In_II this_DD1 present_JJ context_NN1 ,_, suppose_VV0 we_PPIS2 have_VH0 a_AT1 matrix_NN1 A_ZZ1 of_IO order_NN1 n_ZZ1 and_CC require_VV0 to_TO find_VVI its_APPGE reciprocal_JJ ._. 
Then_RT it_PPH1 is_VBZ possible_JJ ,_, using_VVG (_( 1_MC1 )_) ,_, so_RR to_TO choose_VVI ,_, &lsqb;_( formula_NN1 &rsqb;_) ,_, etc._RA that_CST Ao_NP1 is_VBZ transformed_VVN into_II A_ZZ1 ,_, when_CS a_AT1 last_MD simple_JJ calculation_NN1 gives_VVZ &lsqb;_( formula_NN1 &rsqb;_) The_AT method_NN1 has_VHZ many_DA2 variants_NN2 :_: we_PPIS2 can_VM build_VVI A_AT1 column_NN1 by_II column_NN1 ,_, or_CC row_VV0 by_II row_NN1 ,_, or_CC column_NN1 and_CC row_VV0 simultaneously_RR ._. 
We_PPIS2 give_VV0 here_RL what_DDQ seems_VVZ to_TO be_VBI the_AT simplest_JJT approach_NN1 ._. 
We_PPIS2 choose_VV0 &lsqb;_( formula_NN1 &rsqb;_) (_( or_CC possibly_RR a_AT1 scalar_JJ matrix_NN1 &lsqb;_( formula_NN1 &rsqb;_) )_) ._. 
Then_RT ,_, if_CS &lsqb;_( formula_NN1 &rsqb;_) is_VBZ the_AT isolating_JJ row_NN1 vector_NN1 ,_, we_PPIS2 form_VV0 &lsqb;_( formula_NN1 &rsqb;_) Here_RL &lsqb;_( formula_NN1 &rsqb;_) is_VBZ the_AT first_MD column_NN1 of_IO A_ZZ1 except_CS21 that_CS22 the_AT leading_JJ element_NN1 &lsqb;_( formula_NN1 &rsqb;_) is_VBZ diminished_VVN by_II 1_MC1 ;_; thus_RR the_AT first_MD column_NN1 of_IO &lsqb;_( formula_NN1 &rsqb;_) is_VBZ identical_JJ with_IW that_DD1 of_IO A_ZZ1 ,_, while_CS the_AT (_( n_ZZ1 1_MC1 )_) remaining_JJ columns_NN2 are_VBR those_DD2 of_IO I._NP1 By_II Theorem_NN1 IX_MC of_IO 1.22_MC &lsqb;_( formula_NN1 &rsqb;_) Next_MD we_PPIS2 evaluate_VV0 (_( this_DD1 step_NN1 is_VBZ not_XX quite_RG so_RG simple_JJ )_) &lsqb;_( formula_NN1 &rsqb;_) so_CS21 that_CS22 &lsqb;_( formula_NN1 &rsqb;_) Here_RL &lsqb;_( formula_NN1 &rsqb;_) is_VBZ the_AT second_MD column_NN1 of_IO A_ZZ1 but_CCB with_IW the_AT second_MD element_NN1 reduced_VVN by_II 1_MC1 ;_; &lsqb;_( formula_NN1 &rsqb;_) and_CC &lsqb;_( formula_NN1 &rsqb;_) is_VBZ the_AT second_MD element_NN1 of_IO &lsqb;_( formula_NN1 &rsqb;_) Further_JJR steps_NN2 follow_VV0 this_DD1 pattern_NN1 and_CC are_VBR obvious_JJ ._. 
We_PPIS2 may_VM ,_, however_RR ,_, observe_VVI that_CST (_( 4_MC )_) may_VM be_VBI written_VVN &lsqb;_( formula_NN1 &rsqb;_) i.e._REX as_II the_AT sum_NN1 of_IO &lsqb;_( formula_NN1 &rsqb;_) and_CC a_AT1 rank_NN1 1_MC1 matrix_NN1 of_IO which_DDQ the_AT rows_NN2 are_VBR proportional_JJ to_II the_AT second_MD row_NN1 of_IO &lsqb;_( formula_NN1 &rsqb;_) ._. 
Example_NN1 1_MC1 We_PPIS2 choose_VV0 the_AT same_DA numerical_JJ matrix_NN1 as_CSA before_II (_( Equation_NN1 (_( 2.2.1.4_MC )_) )_) ,_, viz_REX :_: &lsqb;_( formula_NN1 &rsqb;_) so_CS21 that_CS22 &lsqb;_( formula_NN1 &rsqb;_) &lsqb;_( formula_NN1 &rsqb;_) 2.2.4_MC Triangulation_NN1 We_PPIS2 have_VH0 already_RR seen_VVN ,_, in_II &lsqb;_( formula_NN1 &rsqb;_) that_DD1 reciprocation_NN1 of_IO a_AT1 triangular_JJ matrix_NN1 can_VM be_VBI achieved_VVN by_II a_AT1 simple_JJ process_NN1 of_IO back-substitution_NN1 ._. 
Accordingly_RR ,_, if_CS we_PPIS2 first_MD reduce_VV0 a_AT1 matrix_NN1 A_ZZ1 to_II triangular_JJ form_NN1 we_PPIS2 can_VM then_RT proceed_VVI to_TO determine_VVI its_APPGE reciprocal_JJ ._. 
The_AT usual_JJ reduction_NN1 to_II triangular_JJ form_NN1 is_VBZ by_II a_AT1 pivotal_JJ condensation_NN1 similar_JJ to_II that_DD1 described_VVN in_II 2.2.1_MC ,_, except_CS21 that_CS22 we_PPIS2 only_RR remove_VV0 elements_NN2 below_II the_AT diagonal_JJ ._. 
We_PPIS2 may_VM best_RRT illustrate_VVI this_DD1 with_IW out_II usual_JJ example_NN1 (_( see_VV0 Table_NN1 1_MC1 )_) ._. 
What_DDQ we_PPIS2 have_VH0 done_VDN is_VBZ to_TO keep_VVI &lsqb;_( formula_NN1 &rsqb;_) but_CCB use_VV0 it_PPH1 to_TO eliminate_VVI the_AT leading_JJ elements_NN2 of_IO &lsqb;_( formula_NN1 &rsqb;_) and_RR31 so_RR32 on_RR33 ,_, reducing_VVG the_AT number_NN1 of_IO rows_NN2 by_II 1_MC1 at_II each_DD1 step_NN1 ._. 
We_PPIS2 have_VH0 now_RT ,_, in_II effect_NN1 ,_, achieved_VVN the_AT equation_NN1 &lsqb;_( formula_NN1 &rsqb;_) This_DD1 has_VHZ been_VBN obtained_VVN by_II triangulation_NN1 without_IW row_NN1 interchange_NN1 ._. 
If_CS we_PPIS2 now_RT proceed_VV0 to_II the_AT process_NN1 of_IO back-substitution_NN1 ,_, this_DD1 is_VBZ in_II effect_NN1 operating_NN1 again_RT with_IW rows_NN2 ,_, and_CC all_DB it_PPH1 does_VDZ is_VBZ to_TO compete_VVI the_AT process_NN1 which_DDQ was_VBDZ done_VDN continuously_RR in_II the_AT pivotal_JJ condensation_NN1 of_IO 2.2.1_MC ._. 
There_EX is_VBZ thus_RR no_AT advantage_NN1 at_II all_DB in_RP he_PPHS1 intermediate_JJ triangulation_NN1 step_NN1 ._. 
However_RR ,_, much_DA1 attention_NN1 has_VHZ been_VBN given_VVN in_II the_AT literature_NN1 to_II inversion_NN1 of_IO triangular_JJ matrices_NN2 by_II methods_NN2 other_II21 than_II22 back-substitution_NN1 ,_, particularly_RR iterative_JJ methods_NN2 ._. 
We_PPIS2 illustrate_VV0 below_RG one_MC1 direct_JJ method_NN1 ._. 
Triangulation_NN1 has_VHZ many_DA2 important_JJ uses_NN2 ._. 
When_CS we_PPIS2 reduce_VV0 a_AT1 matrix_NN1 A_ZZ1 to_II triangular_JJ form_NN1 ,_, we_PPIS2 write_VV0 it_PPH1 as_CSA &lsqb;_( formula_NN1 &rsqb;_) ._. 
We_PPIS2 may_VM first_MD note_VVI that_CST the_AT product_NN1 of_IO the_AT diagonal_JJ elements_NN2 of_IO the_AT leading_JJ matrix_NN1 in_II (_( 1_MC1 )_) is_VBZ evidently_RR &lsqb;_( formula_NN1 &rsqb;_) ._. 
If_CS we_PPIS2 divide_VV0 each_DD1 row_NN1 through_RP by_II its_APPGE diagonal_JJ element_NN1 ,_, (_( 1_MC1 )_) becomes_VVZ &lsqb;_( formula_NN1 &rsqb;_) The_AT leading_JJ matrix_NN1 here_RL is_VBZ of_IO the_AT form_NN1 I_ZZ1 +_FO C_ZZ1 ,_, where_CS C_ZZ1 is_VBZ null_JJ except_II21 for_II22 the_AT elements_NN2 above_II the_AT diagonal_JJ ._. 
Now_RT ,_, provided_VVN &lsqb;_( formula_NN1 &rsqb;_) as_CSA r_ZZ1 increases_VVZ ,_, &lsqb;_( formula_NN1 &rsqb;_) as_CSA may_VM be_VBI checked_VVN by_II premultiplication_NN1 by_II I_ZZ1 +_FO C._ZZ1 In_II this_DD1 example_NN1 &lsqb;_( formula_NN1 &rsqb;_) and_CC on_II use_NN1 of_IO (_( 3_MC )_) we_PPIS2 obtain_VV0 from_II a_AT1 single_JJ multiplication_NN1 &lsqb;_( formula_NN1 &rsqb;_) Premultiplication_NN1 of(2)_FO by_II (_( 4_MC )_) yields_VVZ finally_RR &lsqb;_( formula_NN1 &rsqb;_) in_II agreement_NN1 with_IW results_NN2 given_VVN earlier._NNU 2.3_MC ITERATIVE_JJ METHODS_NN2 These_DD2 methods_NN2 are_VBR well_RR suited_VVN to_II computers_NN2 and_CC often_RR have_VH0 the_AT advantage_NN1 that_CST ,_, should_VM an_AT1 error_NN1 occur_VVI ,_, at_RR21 worst_RR22 it_PPH1 lengthens_VVZ the_AT computations_NN2 a_RR21 little._RR22 2.3.1_MC Improvement_NN1 of_IO an_AT1 approximate_JJ reciprocal_JJ This_DD1 is_VBZ an_AT1 important_JJ topic_NN1 ;_; it_PPH1 often_RR happens_VVZ that_CST an_AT1 approximate_JJ reciprocal_JJ of_IO a_AT1 matrix_NN1 A_ZZ1 is_VBZ known_VVN :_: perhaps_RR from_II a_AT1 rough_JJ calculation_NN1 ,_, or_CC one_PN1 in_II which_DDQ an_AT1 error_NN1 has_VHZ been_VBN made_VVN ,_, or_CC even_RR that_CST belonging_VVG to_II a_AT1 neighbour_NN1 matrix_NN1 of_IO A._NNU Refinement_NN1 of_IO this_DD1 approximate_JJ reciprocal_JJ is_VBZ then_RT required_VVN ._. 
The_AT usual_JJ procedure_NN1 is_VBZ as_CSA follows_VVZ ._. 
Let_VV0 R_ZZ1 be_VBI the_AT exact_JJ reciprocal_JJ and_CC &lsqb;_( formula_NN1 &rsqb;_) the_AT approximation_NN1 ._. 
Write_VV0 &lsqb;_( formula_NN1 &rsqb;_) Then_RT &lsqb;_( formula_NN1 &rsqb;_) Premultiply_RR by_II &lsqb;_( formula_NN1 &rsqb;_) then_RT since_CS &lsqb;_( formula_NN1 &rsqb;_) &lsqb;_( formula_NN1 &rsqb;_) Thus_RR &lsqb;_( formula_NN1 &rsqb;_) is_VBZ a_AT1 new_JJ approximation_NN1 to_II R_ZZ1 ,_, and_CC we_PPIS2 may_VM repeat_VVI the_AT cycle_NN1 as_CSA required_VVN to_TO obtain_VVI R_ZZ1 ,_, subject_II21 to_II22 convergence_NN1 of_IO the_AT method_NN1 ._. 
The_AT conditions_NN2 for_IF this_DD1 are_VBR readily_RR established_VVN ._. 
From_II (_( 2_MC )_) we_PPIS2 find._NNU &lsqb;_( formula_NN1 &rsqb;_) and_CC further_JJR steps_NN2 give_VV0 &lsqb;_( formula_NN1 &rsqb;_) and_RR31 so_RR32 on_RR33 ._. 
Thus_RR ,_, provided_VVN &lsqb;_( formula_NN1 &rsqb;_) as_CSA r_ZZ1 increases_VVZ ,_, the_AT method_NN1 will_VM converge_VVI very_RG rapidly_RR ._. 
This_DD1 will_VM normally_RR be_VBI the_AT case_NN1 of_IO R1_FO is_VBZ a_AT1 reasonably_RR good_JJ approximation_NN1 ,_, when_CS the_AT elements_NN2 of_IO &lsqb;_( formula_NN1 &rsqb;_) will_VM be_VBI small_JJ ._. 
At_II each_DD1 step_NN1 the_AT method_NN1 requires_VVZ the_AT multiplication_NN1 of_IO two_MC matrices_NN2 such_II21 as_II22 AR1_FO ,_, the_AT subtraction_NN1 of_IO the_AT result_NN1 from_II 2I_FO ,_, and_CC then_RT the_AT second_MD multiplication_NN1 ,_, e.g._REX &lsqb;_( formula_NN1 &rsqb;_) to_TO obtain_VVI the_AT next_MD approximation_NN1 ._. 
An_AT1 alternative_JJ procedure_NN1 which_DDQ achieves_VVZ the_AT same_DA result_NN1 but_CCB is_VBZ more_RGR convenient_JJ computationally_RR is_VBZ the_AT following_JJ ._. 
Write_VV0 &lsqb;_( formula_NN1 &rsqb;_) Then_RT &lsqb;_( formula_NN1 &rsqb;_) Inversion_NN1 of_IO this_DD1 and_CC premultiplication_NN1 of_IO the_AT result_NN1 by_II R1_FO gives_VVZ ,_, provided_VVN &lsqb;_( formula_NN1 &rsqb;_) as_CSA r_ZZ1 increases._NNU &lsqb;_( formula_NN1 &rsqb;_) It_PPH1 is_VBZ readily_RR checked_VVN ,_, in_II31 view_II32 of_II33 (_( 3_MC )_) et_NP1 seq._NNU ,_, that_CST he_PPHS1 product_NN1 of_IO the_AT first_MD two_MC terms_NN2 in_II (_( 5_MC )_) is_VBZ R2_FO ,_, of_IO the_AT first_MD three_MC is_VBZ R3_FO ,_, and_RR31 so_RR32 on_RR33 ._. 
The_AT number_NN1 of_IO matrix_NN1 multiplications_NN2 in_II this_DD1 procedure_NN1 is_VBZ the_AT same_DA as_CSA before_RT ;_; but_CCB the_AT squaring_NN1 of_IO &lsqb;_( formula_NN1 &rsqb;_) etc._RA is_VBZ usually_RR relatively_RR easy_JJ since_CS they_PPHS2 become_VV0 rapidly_RR smaller_JJR in_II practice_NN1 ._. 
Indeed_RR ,_, if_CS ,_, for_REX21 example_REX22 in_II (_( 5_MC )_) ,_, &lsqb;_( formula_NN1 &rsqb;_) and_CC higher_JJR powers_NN2 are_VBR sensibly_RR null_JJ to_II the_AT order_NN1 of_IO accuracy_NN1 required_VVN ,_, we_PPIS2 can_VM write_VVI &lsqb;_( formula_NN1 &rsqb;_) saving_VVG one_MC1 multiplication_NN1 ._. 
In_II practice_NN1 ,_, since_CS the_AT numbers_NN2 in_II &lsqb;_( formula_NN1 &rsqb;_) and_CC I_PPIS1 become_VV0 progressively_RR and_CC rapidly_RR more_RGR disparate_JJ ,_, it_PPH1 is_VBZ better_JJR to_TO deal_VVI with_IW them_PPHO2 separately_RR ;_; i.e._REX the_AT operations_NN2 in_II (_( 5_MC )_) are_VBR carried_VVN on_RP as_CSA follows_VVZ ;_; we_PPIS2 are_VBR given_VVN A_ZZ1 and_CC R1_FO :_: (_( i_ZZ1 )_) evaluate_VV0 &lsqb;_( formula_NN1 &rsqb;_) (_( ii_MC )_) evaluation_NN1 &lsqb;_( formula_NN1 &rsqb;_) ..._... until_CS to_II the_AT order_NN1 of_IO accuracy_NN1 required_VVN a_AT1 power_NN1 of_IO E1_FO is_VBZ sensibly_RR null._NNU (_( ii_MC )_) evaluate_VV0 &lsqb;_( formula_NN1 &rsqb;_) and_CC add_VV0 R_ZZ1 1_MC1 to_TO get_VVI R_ZZ1 2_MC ,_, (_( iv_MC )_) evaluate_VV0 &lsqb;_( formula_NN1 &rsqb;_) and_CC add_VV0 &lsqb;_( formula_NN1 &rsqb;_) to_TO get_VVI &lsqb;_( formula_NN1 &rsqb;_) ,_, (_( v_ZZ1 )_) evaluate_VV0 &lsqb;_( formula_NN1 &rsqb;_) and_CC add_VV0 &lsqb;_( formula_NN1 &rsqb;_) to_TO get_VVI &lsqb;_( formula_NN1 &rsqb;_) ,_, and_RR31 so_RR32 on_RR33 ._. 
Normally_RR ,_, very_RG few_DA2 steps_NN2 are_VBR needed_VVN ._. 
Example_NN1 1_MC1 Suppose_VV0 we_PPIS2 are_VBR given_VVN &lsqb;_( formula_NN1 &rsqb;_) Then_RT A_ZZ1 is_VBZ the_AT matrix_NN1 of_IO our_APPGE previous_JJ examples_NN2 ,_, and_CC R1_FO an_AT1 approximate_JJ reciprocal_JJ ,_, given_VVN to_II three_MC places_NN2 of_IO decimals_NN2 ._. 
It_PPH1 is_VBZ required_VVN to_TO find_VVI the_AT reciprocal_JJ R_ZZ1 of_IO A_ZZ1 ,_, correct_JJ to_II six_MC places_NN2 of_IO decimals_NN2 ._. 
Then_RT &lsqb;_( formula_NN1 &rsqb;_) where_RRQ the_AT figures_NN2 are_VBR exact._NNU &lsqb;_( formula_NN1 &rsqb;_) is_VBZ thus_RR exact_JJ to_II six_MC decimal_JJ places_NN2 ,_, viz_REX :_: &lsqb;_( formula_NN1 &rsqb;_) but_CCB &lsqb;_( formula_NN1 &rsqb;_) would_VM require_VVI 12_MC decimal_JJ places_NN2 ._. 
We_PPIS2 retain_VV0 only_RR eight_MC :_: &lsqb;_( formula_NN1 &rsqb;_) an_AT1 to_II this_DD1 order_NN1 of_IO accuracy_NN1 &lsqb;_( formula_NN1 &rsqb;_) is_VBZ null_JJ ._. 
Hence_RR &lsqb;_( formula_NN1 &rsqb;_) Since_CS &lsqb;_( formula_NN1 &rsqb;_) is_VBZ null_JJ ,_, we_PPIS2 can_VM apply_VVI (_( 6_MC )_) ,_, when_RRQ &lsqb;_( formula_NN1 &rsqb;_) where_CS we_PPIS2 have_VH0 retained_VVN eight_MC decimal_JJ places_NN2 ._. 
Finally_RR ,_, to_II six_MC decimal_JJ places_NN2 &lsqb;_( formula_NN1 &rsqb;_) 2.3.1_MC General_JJ Iteration_NN1 The_AT most_RGT general_JJ iteration_NN1 procedure_NN1 appears_VVZ to_TO be_VBI the_AT following_JJ ._. 
We_PPIS2 require_VV0 to_TO solve_VVI RA_NP1 =_FO I_ZZ1 for_IF R._NP1 Write_VV0 the_AT known_JJ matrix_NN1 A_ZZ1 as_CSA A_ZZ1 =_FO B_ZZ1 C_ZZ1 ,_, where_CS B_ZZ1 is_VBZ a_AT1 matrix_NN1 having_VHG a_AT1 known_JJ (_( or_CC easily_RR found_VVN )_) reciprocal_JJ ._. 
Then_RT the_AT problem_NN1 can_VM be_VBI written_VVN &lsqb;_( formula_NN1 &rsqb;_) where_CS we_PPIS2 have_VH0 written_VVN &lsqb;_( formula_NN1 &rsqb;_) There_EX are_VBR various_JJ iterative_JJ methods_NN2 for_IF solving_NN1 (_( 2_MC )_) ._. 
For_REX21 example_REX22 ,_, we_PPIS2 may_VM write_VVI it_PPH1 as_CSA &lsqb;_( formula_NN1 &rsqb;_) and_CC use_VV0 the_AT regression_NN1 formula_NN1 &lsqb;_( formula_NN1 &rsqb;_) so_CS21 that_CS22 ,_, for_REX21 example_REX22 ,_, if_CS Ro_NP1 =_FO 0_MC ,_, we_PPIS2 should_VM obtain_VVI &lsqb;_( formula_NN1 &rsqb;_) and_RR31 so_RR32 on_RR33 ._. 
Evidently_RR ,_, these_DD2 are_VBR successive_JJ steps_NN2 in_II the_AT solution_NN1 of_IO (_( 2_MC )_) ,_, which_DDQ is_VBZ valid_JJ of_IO &lsqb;_( formula_NN1 &rsqb;_) as_CSA r_ZZ1 increases._NNU &lsqb;_( formula_NN1 &rsqb;_) A_AT1 rather_RG more_RGR rapid_JJ method_NN1 is_VBZ to_TO write_VVI (_( 5_MC )_) as_CSA &lsqb;_( formula_NN1 &rsqb;_) For_IF the_AT simple_JJ reciprocation_NN1 of_IO a_AT1 matrix_NN1 A_ZZ1 these_DD2 methods_NN2 hardly_RR compete_VV0 with_IW pivotal_JJ condensation_NN1 ;_; however_RR ,_, they_PPHS2 do_VD0 have_VHI a_AT1 place_NN1 in_II the_AT parallel_JJ problem_NN1 of_IO solution_NN1 of_IO linear_JJ algebraic_JJ equations_NN2 ._. 
It_PPH1 must_VM be_VBI stressed_VVN ,_, however_RR ,_, that_CST the_AT choice_NN1 of_IO B_ZZ1 must_VM be_VBI such_DA at_II &lsqb;_( formula_NN1 &rsqb;_) has_VHZ eigenvalues_NN2 with_IW moduli_NN2 all_RR less_DAR than_CSN unity_NN1 ;_; Sylvester_NP1 's_GE expansion_NN1 then_RT shows_VVZ that_CST &lsqb;_( formula_NN1 &rsqb;_) as_CSA r_ZZ1 increases_VVZ ._. 
There_EX are_VBR certain_JJ physical_JJ problems_NN2 yielding_VVG matrices_NN2 for_IF which_DDQ this_DD1 can_VM be_VBI guaranteed_VVN ;_; but_CCB in_RR21 general_RR22 some_DD trial_NN1 and_CC error_NN1 is_VBZ involved_VVN ._. 
It_PPH1 should_VM be_VBI noted_VVN that_CST (_( 2_MC )_) can_VM be_VBI written_VVN alternatively_RR as_CSA &lsqb;_( formula_NN1 &rsqb;_) leading_VVG to_II &lsqb;_( formula_NN1 &rsqb;_) It_PPH1 is_VBZ important_JJ to_TO observe_VVI that_CST this_DD1 general_JJ iteration_NN1 procedure_NN1 is_VBZ really_RR exactly_RR the_AT same_DA as_CSA the_AT method_NN1 given_VVN in_II 2.3.1_MC for_IF improvement_NN1 of_IO an_AT1 approximate_JJ reciprocal_JJ ._. 
The_AT first_MD of_IO Equations_NN2 (_( 4_MC )_) shows_VVZ that_CST &lsqb;_( formula_NN1 &rsqb;_) is_VBZ an_AT1 approximate_JJ reciprocal_JJ ,_, and_CC then_RT Equations_NN2 (_( 2.3.1.5_MC )_) and_CC (_( 6_MC )_) are_VBR the_AT same_DA ,_, provided_CS E1_FO and_CC E_ZZ1 are_VBR the_AT same_DA ._. 
Now_RT ,_, if_CS B-1_FO is_VBZ the_AT same_DA as_CSA R1_FO ,_, &lsqb;_( formula_NN1 &rsqb;_) so_CS21 that_CS22 the_AT methods_NN2 are_VBR effectively_RR the_AT same_DA ._. 
This_DD1 result_NN1 shows_VVZ that_CST ,_, although_CS in_II (_( 1_MC1 )_) no_AT restriction_NN1 is_VBZ placed_VVN on_II B_ZZ1 ,_, it_PPH1 must_VM in_II practice_NN1 be_VBI a_AT1 reasonable_JJ approximation_NN1 to_II &lsqb;_( formula_NN1 &rsqb;_) if_CS the_AT method_NN1 is_VBZ to_TO converge_VVI ._. 
Example_NN1 1_MC1 We_PPIS2 illustrate_VV0 Equation_NN1 (_( 6_MC )_) here_RL ._. 
Let_VV0 &lsqb;_( formula_NN1 &rsqb;_) Here_RL we_PPIS2 have_VH0 chosen_VVN B_ZZ1 as_II the_AT lower_JJR triangular_JJ portion_NN1 of_IO A_ZZ1 ,_, and_CC C_ZZ1 is_VBZ of_RR21 course_RR22 B_ZZ1 A._NP1 We_PPIS2 are_VBR given_CS21 that_CS22 &lsqb;_( formula_NN1 &rsqb;_) To_II four_MC places_NN2 of_IO decimals_NN2 we_PPIS2 now_RT find_VV0 &lsqb;_( formula_NN1 &rsqb;_) and_CC to_II this_DD1 order_NN1 of_IO accuracy_NN1 ,_, &lsqb;_( formula_NN1 &rsqb;_) is_VBZ null_JJ ._. 
Then_RT ,_, successively_RR (_( note_VV0 that_CST these_DD2 are_VBR not_XX the_AT steps_NN2 of_IO (_( 4_MC )_) )_) ._. 
&lsqb;_( formula_NN1 &rsqb;_) and_CC &lsqb;_( formula_NN1 &rsqb;_) is_VBZ in_II fact_NN1 the_AT exact_JJ reciprocal_JJ of_IO A._NP1 2.3.3_MC Special_JJ iteration_NN1 procedures_NN2 In_II Equation_NN1 (_( 2.3.2.1_MC )_) the_AT only_JJ restriction_NN1 placed_VVN on_II B_NP1 was_VBDZ that_CST it_PPH1 should_VM have_VHI a_AT1 known_JJ or_CC easily-found_JJ reciprocal_JJ ._. 
We_PPIS2 now_RT discuss_VV0 two_MC particular_JJ choices_NN2 for_IF B._NP1 (_( a_ZZ1 )_) Seidel_NP1 iteration_NN1 ._. 
Here_RL B_ZZ1 is_VBZ chosen_VVN to_TO be_VBI the_AT lower_JJR triangular_JJ part_NN1 of_IO A_ZZ1 ,_, including_II the_AT diagonal_JJ ,_, with_IW all_DB superdiagonal_JJ elements_NN2 null_JJ ._. 
B_ZZ1 is_VBZ then_RT reciprocated_VVN as_CSA in_II 1.9(8)_FO an_AT1 the_AT methods_NN2 of_IO the_AT last_MD paragraph_NN1 applied_VVD ._. 
The_AT example_NN1 given_VVN there_RL is_VBZ in_II fact_NN1 an_AT1 illustration_NN1 of_IO Seidel_NP1 iteration._NNU (_( b_ZZ1 )_) Simple_JJ or_CC diagonal_JJ iteration_NN1 ._. 
Here_RL B_ZZ1 is_VBZ chosen_VVN to_TO be_VBI the_AT diagonal_JJ matrix_NN1 of_IO the_AT diagonal_JJ elements_NN2 of_IO A._NP1 Its_APPGE reciprocal_JJ can_VM therefore_RR be_VBI written_VVN down_RP ._. 
It_PPH1 is_VBZ mostly_RR used_VVN when_CS A_ZZ1 is_VBZ a_AT1 &quot;_" sparse_JJ &quot;_" matrix_NN1 ,_, with_IW elements_NN2 dominated_VVN by_II the_AT diagonal_JJ ;_; B-1_FO is_VBZ then_RT a_AT1 reasonable_JJ approximation_NN1 to_II R._NP1 If_CS this_DD1 is_VBZ not_XX the_AT case_NN1 ,_, the_AT method_NN1 may_VM not_XX converge_VVI ,_, or_CC do_VD0 so_RR very_RG slowly_RR ._. 
For_REX21 example_REX22 ,_, if_CS we_PPIS2 take_VV0 the_AT matrix_NN1 A_ZZ1 of_IO the_AT last_MD example_NN1 ,_, an_AT1 choose_VV0 &lsqb;_( formula_NN1 &rsqb;_) then_RT &lsqb;_( formula_NN1 &rsqb;_) This_DD1 matrix_NN1 has_VHZ two_MC real_JJ eigenvalues_NN2 ,_, 0.319_MC and_CC 0.597_MC ,_, and_CC a_AT1 pair_NN of_IO complex_JJ eigenvalues_NN2 of_IO modulus_NN1 0.999_MC ._. 
The_AT powers_NN2 of_IO E_ZZ1 therefore_RR converge_VV0 very_RG slowly_RR indeed_RR ._. 
A_AT1 better_JJR illustration_NN1 follows_VVZ ._. 
Example_NN1 1_MC1 Let_VV0 &lsqb;_( formula_NN1 &rsqb;_) Then_RT ,_, if_CS we_PPIS2 retain_VV0 eight_MC decimal_JJ places_NN2 &lsqb;_( formula_NN1 &rsqb;_) Hence_RR ,_, if_CS &lsqb;_( formula_NN1 &rsqb;_) and_RR31 so_RR32 on_RR33 ,_, we_PPIS2 find_VV0 as_RG successive_JJ approximations_NN2 to_II &lsqb;_( formula_NN1 &rsqb;_) correct_VV0 to_II six_MC decimal_JJ places_NN2 ,_, &lsqb;_( formula_NN1 &rsqb;_) 2.4_MC ._. 
SPECIAL_JJ TYPES_NN2 OF_IO MATRIX_NN1 Certain_JJ matrices_NN2 have_VH0 properties_NN2 which_DDQ make_VV0 reciprocation_NN1 rather_RG easier_RRR than_CSN would_VM otherwise_RR be_VBI the_AT case_NN1 ._. 
We_PPIS2 give_VV0 a_AT1 few_DA2 examples._NNU 2.4.1_MC Symmetric_JJ matrices_NN2 The_AT reciprocal_JJ R_ZZ1 of_IO a_AT1 symmetric_JJ matrix_NN1 is_VBZ itself_PPX1 symmetric_JJ ._. 
For_IF if_CS A_ZZ1 is_VBZ symmetric_JJ and_CC we_PPIS2 transpose_VV0 AR_UH =_FO I_ZZ1 to_TO get_VVI &lsqb;_( formula_NN1 &rsqb;_) then_RT evidently_RR &lsqb;_( formula_NN1 &rsqb;_) (_( see_VV0 also_RR 1.22_MC ,_, Theorem_NN1 I_ZZ1 ,_, Corollary_NN1 )_) ._. 
&lsqb;_( formula_NN1 &rsqb;_) It_PPH1 follows_VVZ that_CST we_PPIS2 do_VD0 not_XX need_VVI to_TO compute_VVI all_DB the_AT &lsqb;_( formula_NN1 &rsqb;_) elements_NN2 of_IO R_ZZ1 ;_; n_ZZ1 (_( n_ZZ1 +_FO 1_MC1 )_) /2_MF elements_NN2 ,_, comprised_VVN in_II the_AT diagonal_JJ and_CC below_RL ,_, are_VBR all_DB that_CST are_VBR required_VVN ._. 
Example_NN1 1_MC1 We_PPIS2 give_VV0 in_RP Table_NN1 1_MC1 a_AT1 scheme_NN1 for_IF pivotal_JJ condensation_NN1 of_IO the_AT matrix_NN1 just_RR examined_VVN in_II Example_NN1 2.3.3.1_MC ._. 
For_IF convenience_NN1 ,_, the_AT reciprocal_JJ has_VHZ been_VBN multiplied_VVN by_II 103_MC ;_; we_PPIS2 work_VV0 to_II six_MC decimal_JJ places_NN2 on_II the_AT left_JJ ,_, three_MC on_II the_AT right_NN1 ._. 
If_CS the_AT last_MD four_MC rows_NN2 of_IO Table_NN1 1_MC1 are_VBR taken_VVN in_II reverse_JJ order_NN1 ,_, the_AT required_JJ reciprocal_JJ is_VBZ obtained_VVN ._. 
It_PPH1 is_VBZ to_TO be_VBI noted_VVN that_CST the_AT starred_JJ numbers_NN2 in_II Table_NN1 1_MC1 do_VD0 not_XX need_VVI to_TO be_VBI calculated_VVN ,_, though_CS it_PPH1 is_VBZ convenient_JJ to_TO write_VVI them_PPHO2 down_RP ._. 
Provided_CS21 that_CS22 appropriate_JJ proportions_NN2 of_IO &lsqb;_( formula_NN1 &rsqb;_) are_VBR subtracted_VVN from_II &lsqb;_( formula_NN1 &rsqb;_) as_CSA they_PPHS2 stand_VV0 to_TO give_VVI zeros_NN2 in_II the_AT first_MD column_NN1 ,_, the_AT array_NN1 of_IO numbers_NN2 on_II the_AT left_JJ in_II &lsqb;_( formula_NN1 &rsqb;_) can_VM be_VBI written_VVN down_RP ;_; when_CS the_AT rest_NN1 of_IO &lsqb;_( formula_NN1 &rsqb;_) is_VBZ found_VVN ,_, the_AT third_MD element_NN1 of_IO &lsqb;_( formula_NN1 &rsqb;_) can_VM be_VBI inserted_VVN ,_, and_RR31 so_RR32 on_RR33 ._. 
Also_RR ,_, when_RRQ &lsqb;_( formula_NN1 &rsqb;_) is_VBZ reached_VVN ,_, this_DD1 gives_VVZ the_AT last_MD row_NN1 of_IO R_ZZ1 and_CC hence_RR also_RR the_AT last_MD column_NN1 ,_, so_CS21 that_CS22 the_AT eighth_MD elements_NN2 in_II &lsqb;_( formula_NN1 &rsqb;_) can_VM be_VBI written_VVN down_RP ,_, and_RR31 so_RR32 on_RR33 ._. 
However_RR ,_, in_II practice_NN1 ,_, the_AT programming_NN1 required_VVN to_TO avoid_VVI these_DD2 calculations_NN2 could_VM be_VBI more_RGR time-consuming_JJ than_CSN the_AT straight_JJ pivotal_JJ condensation_NN1 ,_, when_CS the_AT symmetry_NN1 of_IO the_AT final_JJ result_NN1 gives_VVZ a_AT1 check_NN1 on_II the_AT calculations_NN2 ._. 
The_AT procedure_NN1 in_II Table_NN1 1_MC1 also_RR exemplifies_VVZ triangulation_NN1 without_IW row_NN1 interchange_NN1 ._. 
The_AT product_NN1 of_IO the_AT underlines_VVZ elements_NN2 in_II the_AT table_NN1 gives_VVZ &lsqb;_( formula_NN1 &rsqb;_) ._. 
2.4.2_MC Positive_JJ definite_JJ (_( symmetric_JJ )_) matrix_NN1 The_AT method_NN1 to_TO be_VBI described_VVN is_VBZ usually_RR attributed_VVN to_II Choleski_NP1 ._. 
It_PPH1 is_VBZ a_AT1 simple_JJ matter_NN1 ,_, which_DDQ we_PPIS2 leave_VV0 to_II the_AT reader_NN1 ,_, to_TO show_VVI that_CST if_CS A_ZZ1 is_VBZ real_JJ ,_, symmetric_JJ and_CC positive_JJ definite_JJ ,_, then_RT it_PPH1 can_VM be_VBI written_VVN as_II the_AT product_NN1 BTB_NP1 where_RRQ B_ZZ1 is_VBZ real_JJ and_CC triangular_JJ ._. 
Then_RT if_CS &lsqb;_( formula_NN1 &rsqb;_) and_CC since_CS B_ZZ1 is_VBZ triangular_JJ ,_, &lsqb;_( formula_NN1 &rsqb;_) is_VBZ readily_RR found_VVN ._. 
We_PPIS2 give_VV0 a_AT1 simple_JJ example_NN1 ._. 
Example_NN1 1_MC1 Let_VV0 &lsqb;_( formula_NN1 &rsqb;_) The_AT leading_JJ element_NN1 gives_VVZ &lsqb;_( formula_NN1 &rsqb;_) (_( use_NN1 of_IO 10_MC makes_VVZ no_AT difference_NN1 )_) ._. 
Then_RT ab_VV0 =_FO 30_MC ,_, b_ZZ1 =_FO 3_MC ;_; ac_NN1 =_FO 10_MC ,_, c_ZZ1 =_FO 1_MC1 ;_; ad_NN1 =_FO -10_MC ,_, d_ZZ1 =_FO -1_MC ._. 
Next_MD &lsqb;_( formula_NN1 &rsqb;_) and_RR31 so_RR32 on_RR33 ._. 
In_II this_DD1 way_NN1 it_PPH1 is_VBZ quickly_RR established_VVN that_CST &lsqb;_( formula_NN1 &rsqb;_) Inversion_NN1 of_IO B_ZZ1 as_CSA in_II 1.9(8)_FO then_RT yields_VVZ &lsqb;_( formula_NN1 &rsqb;_) and_CC then_RT &lsqb;_( formula_NN1 &rsqb;_) It_PPH1 may_VM be_VBI of_IO interest_NN1 to_TO note_VVI that_CST if_CS x_ZZ1 =_FO p_ZZ1 ,_, q_ZZ1 ,_, r_ZZ1 ,_, s_ZZ1 is_VBZ any_DD vector_NN1 of_IO real_JJ numbers_NN2 ,_, not_XX all_DB zero_MC ,_, then_RT the_AT quadratic_JJ form_NN1 &lsqb;_( formula_NN1 &rsqb;_) may_VM be_VBI written_VVN (_( see_VV0 also_RR 1.22_MC ,_, Theorem_NN1 XIII_MC )_) &lsqb;_( formula_NN1 &rsqb;_) where_RRQ &lsqb;_( formula_NN1 &rsqb;_) so_CS21 that_CS22 ,_, where_CS the_AT second_MD term_NN1 does_VDZ not_XX involve_VVI p_ZZ1 ,_, the_AT third_MD p_ZZ1 ,_, q_ZZ1 ,_, and_CC the_AT fourth_MD p_ZZ1 ,_, q_ZZ1 ,_, r_ZZ1 ,_, &lsqb;_( formula_NN1 &rsqb;_) and_CC is_VBZ always_RR positive_JJ under_II the_AT conditions_NN2 given_VVN ._. 
If_CS A_ZZ1 is_VBZ not_XX pos._NNU def._NNU ,_, the_AT elements_NN2 of_IO B_ZZ1 are_VBR in_RR21 general_RR22 complex._NNU 2.4.3_MC Persymmetric_JJ matrix_NN1 The_AT reciprocal_JJ of_IO a_AT1 persymmetric_JJ matrix_NN1 is_VBZ not_XX in_RR21 general_RR22 also_RR persymmetric_JJ ;_; e.g._REX &lsqb;_( formula_NN1 &rsqb;_) A_ZZ1 persymmetric_JJ matrix_NN1 can_VM be_VBI treated_VVN as_CSA symmetric_JJ for_IF the_AT purpose_NN1 of_IO reciprocation_NN1 ,_, but_CCB otherwise_RR it_PPH1 is_VBZ not_XX special._NNU 2.4.4_MC Centrosymmetric_JJ matrix_NN1 We_PPIS2 shall_VM consider_VVI here_RL only_JJ matrices_NN2 of_IO even_JJ order_NN1 ;_; the_AT odd-order_JJ case_NN1 is_VBZ quite_RG straightforward_JJ ,_, but_CCB is_VBZ algebraically_RR more_RGR complicated_JJ ._. 
A_AT1 centrosymmetric_JJ matrix_NN1 A_ZZ1 is_VBZ characterise_VV0 by_II A_ZZ1 =_FO JAJ_NP1 ,_, where_CS J_ZZ1 is_VBZ the_AT reversing_NN1 matrix_NN1 ._. 
Inversion_NN1 of_IO this_DD1 gives_VVZ &lsqb;_( formula_NN1 &rsqb;_) since_CS J2_FO =_FO I_ZZ1 ;_; the_AT reciprocal_JJ of_IO A_ZZ1 is_VBZ therefore_RR also_RR centrosymmetric_JJ ,_, as_CSA are_VBR the_AT integral_JJ powers_NN2 of_IO A._NNU If_CS A_ZZ1 is_VBZ of_IO order_NN1 2m_NNU ,_, it_PPH1 can_VM be_VBI partitioned_VVN into_II four_MC submatrices_NN2 ,_, each_DD1 of_IO order_NN1 m_ZZ1 ,_, thus_RR &lsqb;_( formula_NN1 &rsqb;_) other_JJ representations_NN2 are_VBR possible_JJ ,_, but_CCB this_DD1 appears_VVZ to_TO be_VBI the_AT most_RGT convenient_JJ ._. 
If_CS (_( 2_MC )_) is_VBZ premultiplied_VVN and_CC postmultiplied_VVN by_II &lsqb;_( formula_NN1 &rsqb;_) the_AT result_NN1 is_VBZ A_ZZ1 as_CSA required_VVN ._. 
We_PPIS2 can_VM write_VVI the_AT reciprocal_JJ of_IO A_ZZ1 as_CSA &lsqb;_( formula_NN1 &rsqb;_) and_CC then_RT the_AT product_NN1 of_IO (_( 2_MC )_) and_CC (_( 3_MC )_) yields_NN2 (_( twice_RR )_) &lsqb;_( formula_NN1 &rsqb;_) These_DD2 may_VM be_VBI solved_VVN if_CS either_RR P_ZZ1 or_CC Q_ZZ1 is_VBZ non-singular_JJ ;_; we_PPIS2 assume_VV0 P-1_FO exists_VVZ ,_, and_CC we_PPIS2 write_VV0 R_ZZ1 for_IF P-1Q_FO ._. 
Then_RT from_II (_( 4_MC )_) we_PPIS2 obtain_VV0 &lsqb;_( formula_NN1 &rsqb;_) The_AT reciprocation_NN1 of_IO A_ZZ1 may_VM thus_RR be_VBI effected_VVN through_II two_MC reciprocations_NN2 of_IO order_NN1 m_ZZ1 ,_, instead_II21 of_II22 the_AT much_RR more_RGR laborious_JJ single_JJ reciprocation_NN1 of_IO order_NN1 2m_NNU ._. 
Example_NN1 1_MC1 Consider_VV0 &lsqb;_( formula_NN1 &rsqb;_) Then_RT ,_, in_II the_AT above_JJ nomenclature_NN1 &lsqb;_( formula_NN1 &rsqb;_) Accordingly_RR ,_, &lsqb;_( formula_NN1 &rsqb;_) A_ZZ1 caution_NN1 must_VM be_VBI added_VVN :_: the_AT method_NN1 is_VBZ not_XX always_RR viable_JJ ._. 
Thus_RR ,_, for_REX21 example_REX22 ,_, &lsqb;_( formula_NN1 &rsqb;_) but_CCB this_DD1 result_NN1 can_VM not_XX be_VBI obtained_VVN by_II use_NN1 of_IO (_( 4_MC )_) since_CS by_II inspection_NN1 the_AT submatrices_NN2 of_IO order_NN1 2_MC are_VBR all_RR singular._NNU 2.5_MC LINEAR_JJ ALGEBRAIC_JJ EQUATIONS_NN2 A_ZZ1 set_NN1 of_IO linear_JJ algebraic_JJ equations_NN2 (_( often_RR called_VVN simultaneous_JJ equations_NN2 )_) has_VHZ the_AT form_NN1 Ax_NP1 =_FO p_ZZ1 ._. 
Here_RL A_ZZ1 is_VBZ ,_, in_RR21 general_RR22 ,_, a_AT1 rectangular_JJ matrix_NN1 of_IO order_NN1 m_ZZ1 n_ZZ1 with_IW given_JJ elements_NN2 ,_, p_ZZ1 is_VBZ a_AT1 given_JJ column_NN1 of_IO m_ZZ1 elements_NN2 ,_, and_CC x_II a_AT1 column_NN1 of_IO n_ZZ1 unknowns_NN2 ;_; it_PPH1 is_VBZ required_VVN to_TO find_VVI x_ZZ1 ._. 
Suppose_VV0 first_MD that_DD1 &lsqb;_( formula_NN1 &rsqb;_) ._. 
Then_RT we_PPIS2 have_VH0 more_DAR equations_NN2 than_CSN unknowns_NN2 ._. 
If_CS (_( 1_MC1 )_) is_VBZ soluble_JJ ,_, there_EX must_VM be_VBI n_ZZ1 independent_JJ equations_NN2 ;_; if_CS these_DD2 are_VBR written_VVN first_MD ,_, we_PPIS2 can_VM partition_VVI (_( 1_MC1 )_) in_II the_AT form_NN1 &lsqb;_( formula_NN1 &rsqb;_) where_RRQ B_ZZ1 is_VBZ square_JJ and_CC non-singular_NN1 ._. 
Then_RT &lsqb;_( formula_NN1 &rsqb;_) If_CS s_ZZ1 does_VDZ not_XX satisfy_VVI (_( 2_MC )_) the_AT set_NN1 of_IO equations_NN2 is_VBZ inconsistent_JJ ,_, and_CC (_( 1_MC1 )_) does_VDZ not_XX have_VHI a_AT1 valid_JJ solution_NN1 ._. 
If_CS ,_, however_RR ,_, the_AT equations_NN2 are_VBR consistent_JJ ,_, then_RT the_AT solution_NN1 (_( 2_MC )_) for_IF x_ZZ1 uses_VVZ only_RR n_ZZ1 of_IO them_PPHO2 ;_; the_AT remainder_NN1 are_VBR superfluous_JJ ._. 
Now_RT let&lsqb;formula&rsqb;_VV0 ._. 
In_II this_DD1 case_NN1 ,_, we_PPIS2 have_VH0 fewer_DAR equations_NN2 than_CSN unknowns_NN2 ;_; we_PPIS2 now_RT partition_VV0 (_( 1_MC1 )_) in_II the_AT form_NN1 (_( B_ZZ1 ,_, C_ZZ1 )_) y_ZZ1 ,_, z_ZZ1 =_FO p_ZZ1 ,_, where_CS B_ZZ1 is_VBZ square_JJ ,_, of_IO order_NN1 m_ZZ1 ,_, and_CC y_ZZ1 is_VBZ a_AT1 column_NN1 of_IO m_ZZ1 quantities_NN2 ._. 
We_PPIS2 now_RT obtain_VV0 the_AT partial_JJ solution_NN1 for_IF x_ZZ1 &lsqb;_( formula_NN1 &rsqb;_) in_II which_DDQ we_PPIS2 determine_VV0 m_ZZ1 of_IO the_AT unknowns_NN2 x_ZZ1 in_II31 terms_II32 of_II33 B_ZZ1 ,_, C_ZZ1 ,_, p_ZZ1 and_CC the_AT remaining_JJ m_ZZ1 n_ZZ1 unknowns_NN2 z_ZZ1 ._. 
This_DD1 is_VBZ the_AT parametric_JJ solution_NN1 of_IO 1.10_MC ._. 
Example_NN1 1_MC1 &lsqb;_( formula_NN1 &rsqb;_) Consider_VV0 as_CSA partitioned_VVN ,_, the_AT solution_NN1 is_VBZ (_( see_VV0 (_( 2_MC )_) )_) &lsqb;_( formula_NN1 &rsqb;_) and_CC &lsqb;_( formula_NN1 &rsqb;_) so_CS21 that_CS22 the_AT equations_NN2 are_VBR consistent_JJ ;_; if_CS they_PPHS2 are_VBR taken_VVN any_DD two_MC at_II a_AT1 time_NNT1 ,_, the_AT three_MC cases_NN2 all_DB yield_VV0 the_AT same_DA answer_NN1 ._. 
However_RR ,_, had_VHD the_AT last_MD element_NN1 in_II p_NN1 been_VBN other_II21 than_II22 4_MC ,_, the_AT three_MC pairs_NN2 of_IO equations_NN2 would_VM have_VHI given_VVN three_MC different_JJ answers_NN2 ._. 
Example_NN1 2_MC &lsqb;_( formula_NN1 &rsqb;_) ._. 
Here_RL ,_, a_AT1 simple_JJ example_NN1 is_VBZ &lsqb;_( formula_NN1 &rsqb;_) which_DDQ yields_VVZ &lsqb;_( formula_NN1 &rsqb;_) In_II Examples_NN2 1_MC1 and_CC 2_MC the_AT main_JJ numerical_JJ problem_NN1 is_VBZ the_AT solution_NN1 of_IO BX_NP1 =_FO r_ZZ1 ,_, By_II =_FO p_ZZ1 Cz_NP1 ,_, where_CS B_ZZ1 is_VBZ square_JJ ._. 
We_PPIS2 now_RT confine_VV0 our_APPGE attention_NN1 to_II the_AT case_NN1 where_CS A_ZZ1 is_VBZ square_JJ and_CC non-singular_NN1 ;_; this_DD1 is_VBZ obviously_RR germane_JJ to_II the_AT above_JJ solutions._NNU 2.5.1_MC Direct_JJ Method_NN1 Our_APPGE problem_NN1 is_VBZ to_TO solve_VVI Ax_NP1 =_FO p_ZZ1 ,_, for_IF x_ZZ1 ;_; here_RL A_ZZ1 is_VBZ a_AT1 given_JJ square_NN1 and_CC non-singular_JJ matrix_NN1 of_IO order_NN1 n_ZZ1 ,_, p_ZZ1 is_VBZ a_AT1 given_JJ column_NN1 of_IO n_ZZ1 quantities_NN2 ,_, and_CC x_II a_AT1 column_NN1 of_IO n_ZZ1 unknowns_NN2 ._. 
Formally_RR ,_, the_AT solution_NN1 is_VBZ obvious_JJ :_: x_ZZ1 =_FO A-1p_NNU ._. 
We_PPIS2 can_VM therefore_RR obtain_VVI our_APPGE solution_NN1 by_II inverting_VVG ,_, using_VVG any_DD of_IO the_AT methods_NN2 of_IO 2.2_MC ,_, 2.3_MC ,_, and_CC postmultiplying_VVG the_AT inverse_NN1 by_II p_ZZ1 ._. 
In_II some_DD problems_NN2 we_PPIS2 require_VV0 to_TO obtain_VVI x_ZZ1 for_IF each_DD1 of_IO a_AT1 set_NN1 of_IO values_NN2 for_IF p_ZZ1 ;_; e.g._REX the_AT elements_NN2 of_IO p_ZZ1 may_VM be_VBI functions_NN2 of_IO an_AT1 independent_JJ variable_NN1 ,_, and_CC we_PPIS2 require_VV0 x_ZZ1 for_IF each_DD1 of_IO a_AT1 number_NN1 of_IO values_NN2 of_IO the_AT variable_NN1 ._. 
If_CS this_DD1 number_NN1 is_VBZ high_RR in_RR21 particular_RR22 if_CS it_PPH1 exceeds_VVZ n_ZZ1 then_JJ labour_NN1 is_VBZ saved_VVN by_II inverting_VVG A_ZZ1 and_CC using_VVG (_( 2_MC )_) ._. 
If_CS ,_, on_II the_AT other_JJ hand_NN1 ,_, the_AT number_NN1 is_VBZ small_JJ ,_, labour_NN1 is_VBZ saved_VVN as_CSA follows_VVZ ._. 
We_PPIS2 use_VV0 the_AT approach_NN1 of_IO 2.2.1_MC ;_; but_CCB instead_II21 of_II22 operating_VVG on_II (_( 2.2.1.1_MC )_) with_IW the_AT matrices_NN2 Mi_NP1 ,_, we_PPIS2 operate_VV0 directly_RR on_II (_( 1_MC1 )_) ._. 
Evidently_RR ,_, we_PPIS2 then_RT progressively_RR approach_VV0 the_AT solution_NN1 (_( 2_MC )_) ._. 
The_AT computations_NN2 ,_, by_II simple_JJ operations_NN2 on_II rows_NN2 ,_, are_VBR effected_VVN in_II exactly_RR the_AT same_DA way_NN1 ._. 
Indeed_RR ,_, reciprocation_NN1 is_VBZ really_RR only_RR the_AT solution_NN1 of_IO a_AT1 particular_JJ set_NN1 of_IO linear_JJ algebraic_JJ equations_NN2 ,_, in_II which_DDQ the_AT vectors_NN2 p_ZZ1 are_VBR successively_RR &lsqb;_( formula_NN1 &rsqb;_) For_IF a_AT1 single_JJ set_NN1 ,_, in_II pace_NN1 of_IO e1_FO we_PPIS2 use_VV0 the_AT single_JJ vector_NN1 p_ZZ1 ._. 
Example_NN1 1_MC1 Suppose_VV0 we_PPIS2 require_VV0 to_TO solve_VVI &lsqb;_( formula_NN1 &rsqb;_) The_AT square_JJ matrix_NN1 used_VVD here_RL is_VBZ that_CST employed_VVD for_IF the_AT Examples_NN2 of_IO 2.2_MC ;_; we_PPIS2 choose_VV0 diagonal_JJ pivoting_JJ without_IW row_NN1 interchange_NN1 ._. 
The_AT working_NN1 of_IO the_AT left-hand_JJ array_NN1 is_VBZ thus_RR identical_JJ with_IW that_DD1 of_IO Example_NN1 2.2.2.1_MC ._. 
See_VV0 Table_NN1 1_MC1 ._. 
The_AT last_MD four_MC figures_NN2 in_II the_AT right-hand_JJ array_NN1 are_VBR given_VVN correct_JJ to_II three_MC decimal_JJ places_NN2 and_CC are_VBR in_II fact_NN1 the_AT exact_JJ solution._NNU 2.5.2_MC Iterative_JJ methods_NN2 We_PPIS2 exemplify_VV0 the_AT general_JJ iteration_NN1 procedure_NN1 of_IO 2.3.2_MC ;_; we_PPIS2 write_VV0 B_ZZ1 C_ZZ1 for_IF A_ZZ1 ,_, where_CS the_AT reciprocal_JJ of_IO B_ZZ1 is_VBZ known_VVN or_CC easily_RR obtained_VVN ._. 
Then_RT Equation_NN1 (_( 2.5.1.1_MC )_) can_VM be_VBI written_VVN &lsqb;_( formula_NN1 &rsqb;_) so_CS21 that_CS22 ,_, if_CS F_ZZ1 is_VBZ written_VVN for_IF B-1C_FO ,_, the_AT solution_NN1 is_VBZ given_VVN by_II &lsqb;_( formula_NN1 &rsqb;_) say_VV0 ._. 
As_CSA before_RT ,_, this_DD1 may_VM be_VBI developed_VVN in_II either_DD1 of_IO two_MC ways_NN2 ,_, provided_VVN &lsqb;_( formula_NN1 &rsqb;_) as_CSA r_ZZ1 increases_VVZ :_: x_ZZ1 =_FO q_ZZ1 +_FO Fx_NN1 ,_, suggesting_VVG the_AT regression_NN1 formula_NN1 &lsqb;_( formula_NN1 &rsqb;_) or_CC &lsqb;_( formula_NN1 &rsqb;_) Though_CS the_AT computations_NN2 may_VM be_VBI different_JJ ,_, these_DD2 two_MC formulae_NN2 are_VBR equivalent_JJ ;_; for_IF if_CS xo_NN1 =_FO 0_MC the_AT regression_NN1 formula_NN1 gives_VVZ &lsqb;_( formula_NN1 &rsqb;_) and_RR31 so_RR32 on_RR33 ._. 
We_PPIS2 illustrate_VV0 (_( 3_MC )_) below_RL ._. 
Example_NN1 1_MC1 It_PPH1 is_VBZ required_VVN to_TO solve_VVI &lsqb;_( formula_NN1 &rsqb;_) The_AT square_JJ matrix_NN1 has_VHZ been_VBN used_VVN in_II Example_NN1 2.3.2.1_MC ._. 
We_PPIS2 again_RT choose_VV0 B_ZZ1 to_TO be_VBI A_AT1 except_CS21 that_CS22 the_AT elements_NN2 to_II the_AT right_NN1 of_IO the_AT principal_JJ diagonal_JJ are_VBR all_DB zero_MC :_: then_RT ,_, as_CSA before_II &lsqb;_( formula_NN1 &rsqb;_) &lsqb;_( formula_NN1 &rsqb;_) Note_VV0 that_CST here_RL F_ZZ1 =_FO B-1C_FO :_: in_II the_AT previous_JJ example_NN1 ,_, E_ZZ1 =_FO CB-1_MC1 ._. 
We_PPIS2 now_RT construct_VV0 the_AT iteration_NN1 table_NN1 (_( see_VV0 Table_NN1 1_MC1 )_) ,_, beginning_VVG with_IW x1_FO =_FO q_ZZ1 and_CC applying_VVG (_( 3_MC )_) ._. 
Since_CS ,_, in_II31 view_II32 of_II33 the_AT fact_NN1 that_CST F_ZZ1 has_VHZ two_MC null_JJ columns_NN2 ,_, the_AT first_MD two_MC elements_NN2 of_IO x_ZZ1 are_VBR always_RR multiplied_VVN by_II zeros_NN2 ,_, it_PPH1 is_VBZ not_XX necessary_JJ to_TO compute_VVI them_PPHO2 until_CS the_AT iterations_NN2 are_VBR complete_JJ ._. 
At_II the_AT beginning_NN1 of_IO the_AT iteration_NN1 ,_, we_PPIS2 have_VH0 retained_VVN only_RR two_MC decimal_JJ places_NN2 :_: later_JJR three_MC ,_, then_RT four_MC ,_, until_CS to_II this_DD1 order_NN1 of_IO accuracy_NN1 the_AT iteration_NN1 repeats_NN2 ._. 
Finally_RR ,_, the_AT first_MD two_MC elements_NN2 are_VBR computed_VVN ,_, when_CS the_AT last_MD column_NN1 (_( here_RL x10_FO )_) is_VBZ the_AT solution_NN1 required_VVN ._. 
It_PPH1 may_VM be_VBI observed_VVN that_CST equations_NN2 of_IO the_AT form_NN1 x_ZZ1 =_FO q_ZZ1 +_FO Fx_NN1 arise_VV0 naturally_RR in_II some_DD dynamical_JJ or_CC quasi-static_JJ problems_NN2 ,_, e.g._REX the_AT distortion_NN1 of_IO a_AT1 structure_NN1 under_II applied_JJ load_NN1 ,_, when_CS the_AT load_NN1 is_VBZ varied_VVN by_II the_AT distortion._NNU 2.5.3_MC Choleski_NP1 's_GE method_NN1 for_IF positive_JJ definite_JJ matrices_NN2 Let_VV0 us_PPIO2 now_RT suppose_VVI that_CST in_II 2.5.1_MC A_ZZ1 is_VBZ pos._NNU def_NN1 ._. 
Then_RT as_CSA in_II 2.4.2_MC we_PPIS2 can_VM write_VVI A_ZZ1 as_CSA &lsqb;_( formula_NN1 &rsqb;_) where_RRQ B_ZZ1 is_VBZ triangular_JJ ._. 
The_AT equations_NN2 for_IF solution_NN1 are_VBR now_RT &lsqb;_( formula_NN1 &rsqb;_) Write_VV0 y_ZZ1 for_IF Bx_NP1 ;_; then_RT (_( 1_MC1 )_) can_VM be_VBI solved_VVN in_II two_MC steps_NN2 :_: &lsqb;_( formula_NN1 &rsqb;_) and_CC then_RT Bx_NP1 =_FO y_ZZ1 for_IF x_ZZ1 ._. 
It_PPH1 may_VM be_VBI noted_VVN that_CST a_AT1 set_NN1 of_IO equations_NN2 Ax_NP1 =_FO p_ZZ1 ,_, where_CS A_ZZ1 is_VBZ general_JJ ,_, may_VM be_VBI cast_VVN in_II the_AT above_JJ form_NN1 by_II premultiplication_NN1 by_II AT_II :_: &lsqb;_( formula_NN1 &rsqb;_) However_RR ,_, this_DD1 does_VDZ not_XX appear_VVI to_TO offer_VVI any_DD computational_JJ advantage_NN1 ._. 
Resolution_NN1 of_IO &lsqb;_( formula_NN1 &rsqb;_) into_II &lsqb;_( formula_NN1 &rsqb;_) implies_VVZ A_ZZ1 =_FO CB_FO where_RRQ C_ZZ1 is_VBZ an_AT1 orthogonal_JJ matrix_NN1 ,_, so_CS21 that_CS22 &lsqb;_( formula_NN1 &rsqb;_) ._. 
Again_RT ,_, however_RR ,_, this_DD1 does_VDZ not_XX appear_VVI to_TO offer_VVI any_DD worthwhile_JJ computational_JJ short-cut_NN1 ._. 
Example_NN1 1_MC1 Suppose_VV0 we_PPIS2 wish_VV0 to_TO solve_VVI &lsqb;_( formula_NN1 &rsqb;_) As_CSA in_II 2.4.2_MC ,_, we_PPIS2 can_VM quickly_RR resolve_VVI the_AT square_JJ matrix_NN1 into_II &lsqb;_( formula_NN1 &rsqb;_) A_ZZ1 self-explanatory_JJ scheme_NN1 for_IF computation_NN1 is_VBZ shown_VVN in_II Table_NN1 1_MC1 ._. 
Here_RL BT_NP1 ,_, p_ZZ1 are_VBR written_VVN at_II the_AT head_NN1 of_IO the_AT left-hand_JJ array_NN1 ;_; by_II obvious_JJ row_NN1 combinations_NN2 we_PPIS2 find_VV0 y_ZZ1 at_II the_AT bottom_NN1 ._. 
This_DD1 and_CC B_ZZ1 are_VBR inserted_VVN at_II the_AT top_NN1 of_IO the_AT right-hand_JJ array_NN1 ,_, when_RRQ again_RT obvious_JJ row_NN1 combinations_NN2 give_VV0 the_AT required_JJ solution_NN1 at_II the_AT bottom_JJ right_NN1 :_: in_II summary_NN1 &lsqb;_( formula_NN1 &rsqb;_) 2.5.4_MC Least_DAT squares_NN2 In_II 2.5_MC we_PPIS2 noted_VVD that_CST if_CS we_PPIS2 have_VH0 more_DAR equations_NN2 than_CSN unknowns_NN2 ,_, the_AT equations_NN2 must_VM be_VBI consistent_JJ ,_, or_CC a_AT1 unique_JJ solution_NN1 does_VDZ not_XX exist_VVI ._. 
Nevertheless_RR ,_, especially_RR in_II experimental_JJ work_NN1 ,_, cases_NN2 often_RR arise_VV0 where_RRQ the_AT number_NN1 of_IO equations_NN2 exceeds_VVZ the_AT number_NN1 of_IO unknowns_NN2 ._. 
For_REX21 example_REX22 ,_, suppose_VV0 we_PPIS2 have_VH0 a_AT1 measurable_JJ quantity_NN1 f_ZZ1 which_DDQ we_PPIS2 have_VH0 reason_NN1 to_TO suppose_VVI is_VBZ a_AT1 polynomial_NN1 function_NN1 of_IO an_AT1 independent_JJ variable_NN1 t_ZZ1 :_: &lsqb;_( formula_NN1 &rsqb;_) and_CC we_PPIS2 wish_VV0 to_TO find_VVI the_AT m_MC coefficients_NN2 &lsqb;_( formula_NN1 &rsqb;_) ._. 
Measurement_NN1 of_IO f_ZZ1 at_II each_DD1 of_IO m_ZZ1 values_NN2 of_IO t_ZZ1 would_VM provide_VVI m_ZZ1 equations_NN2 for_IF the_AT m_MC unknowns_NN2 ;_; however_RR ,_, since_CS f_ZZ1 is_VBZ measured_VVN and_CC so_RR is_VBZ inexact_JJ ,_, it_PPH1 may_VM be_VBI thought_VVN better_RRR to_TO measure_VVI f_ZZ1 at_II n_ZZ1 values_NN2 of_IO t_ZZ1 ,_, where_RRQ &lsqb;_( formula_NN1 &rsqb;_) in_BCL21 order_BCL22 to_TO minimise_VVI the_AT effects_NN2 of_IO experimental_JJ error_NN1 ._. 
How_RRQ do_VD0 we_PPIS2 proceed_VVI to_TO find_VVI optimum_JJ values_NN2 of_IO the_AT coefficients_NN2 &lsqb;_( formula_NN1 &rsqb;_) ?_? 
The_AT &quot;_" least_DAT squares_NN2 &quot;_" solution_NN1 of_IO this_DD1 problem_NN1 is_VBZ due_II21 to_II22 Gauss_NP1 ._. 
Let_VV0 the_AT n_ZZ1 equation_NN1 in_II m_MC unknowns_NN2 &lsqb;_( formula_NN1 &rsqb;_) be_VBI written_VVN Ax_NP1 -_- p_ZZ1 =_FO e_ZZ1 ,_, where_CS A_ZZ1 is_VBZ of_IO the_AT order_NN1 (_( n_ZZ1 m_ZZ1 )_) ,_, x_ZZ1 is_VBZ the_AT column_NN1 of_IO m_ZZ1 unknowns_NN2 &lsqb;_( formula_NN1 &rsqb;_) ,_, p_ZZ1 is_VBZ the_AT column_NN1 of_IO n_ZZ1 measures_VVZ values_NN2 of_IO f_ZZ1 ,_, and_CC e_ZZ1 is_VBZ a_AT1 column_NN1 of_IO n_ZZ1 errors_NN2 ._. 
We_PPIS2 are_VBR given_VVN A_ZZ1 and_CC p_ZZ1 ;_; e_ZZ1 will_VM vary_VVI with_IW x_ZZ1 ._. 
Gauss_NP1 '_GE proposition_NN1 is_VBZ that_CST x_ZZ1 will_VM have_VHI its_APPGE optimum_JJ value_NN1 when_CS the_AT sum_NN1 of_IO the_AT squares_NN2 of_IO the_AT errors_NN2 ,_, ie.e_NNU &lsqb;_( formula_NN1 &rsqb;_) is_VBZ a_AT1 minimum_NN1 ._. 
Now_RT &lsqb;_( formula_NN1 &rsqb;_) and_CC for_IF this_DD1 to_TO be_VBI a_AT1 minimum_NN1 ,_, its_APPGE differential_NN1 with_II31 respect_II32 to_II33 x_ZZ1 (_( see_VV0 1.11_MC )_) must_VM vanish_VVI ;_; i.e._REX &lsqb;_( formula_NN1 &rsqb;_) which_DDQ is_VBZ our_APPGE original_JJ set_NN1 of_IO Equations_NN2 (_( 2_MC )_) with_IW e_ZZ1 =_FO 0_MC ,_, premultiplied_VVN by_RP AT_II ;_; it_PPH1 gives_VVZ m_MC equations_NN2 for_IF the_AT m_MC unknowns_NN2 ._. 
Example_NN1 1_MC1 Suppose_VV0 we_PPIS2 expect_VV0 the_AT relation_NN1 (_( 1_MC1 )_) to_TO be_VBI &lsqb;_( formula_NN1 &rsqb;_) and_CC suppose_VV0 further_RRR that_CST we_PPIS2 have_VH0 the_AT following_JJ experimental_JJ table_NN1 :_: &lsqb;_( formula_NN1 &rsqb;_) Then_RT in_II31 view_II32 of_II33 (_( 5_MC )_) our_APPGE equations_NN2 for_IF solutions_NN2 may_VM be_VBI written_VVN &lsqb;_( formula_NN1 &rsqb;_) Before_CS we_PPIS2 proceed_VV0 to_II the_AT least_DAT squares_NN2 solution_NN1 ,_, let_VV0 us_PPIO2 see_VVI what_DDQ results_NN2 would_VM be_VBI obtained_VVN if_CS we_PPIS2 solved_VVD these_DD2 equations_NN2 three_MC at_II a_AT1 time_NNT1 ,_, with_IW e_ZZ1 taken_VVN to_TO be_VBI zero_NN1 for_IF each_DD1 set_NN1 of_IO three_MC ._. 
We_PPIS2 should_VM get_VVI &lsqb;_( formula_NN1 &rsqb;_) These_DD2 are_VBR all_DB very_RG different_JJ ._. 
Moreover_RR ,_, if_CS we_PPIS2 adopted_VVD the_AT first_MD solution_NN1 and_CC applied_VVD it_PPH1 throughout_RL ,_, e_ZZ1 would_VM become_VVI (_( the_AT first_MD three_MC elements_NN2 have_VH0 been_VBN assumed_VVN zero_MC )_) &lsqb;_( formula_NN1 &rsqb;_) which_DDQ is_VBZ clearly_RR quite_RG unacceptable_JJ for_IF the_AT later_JJR measurements_NN2 ._. 
However_RR ,_, if_CS we_PPIS2 premultiply_RR (_( 6_MC )_) by_II AT_II as_CSA in_II (_( 4_MC )_) we_PPIS2 get_VV0 &lsqb;_( formula_NN1 &rsqb;_) and_CC the_AT solution_NN1 ,_, by_II inspection_NN1 ,_, is_VBZ x_ZZ1 =_FO 1_MC1 ,_, -2,1_MC ,_, i.e._REX &lsqb;_( formula_NN1 &rsqb;_) If_CS we_PPIS2 now_RT return_VV0 to_II (_( 6_MC )_) and_CC use_VV0 this_DD1 solution_NN1 ,_, we_PPIS2 find_VV0 &lsqb;_( formula_NN1 &rsqb;_) which_DDQ compares_VVZ very_RG favourably_RR with_IW the_AT result_NN1 given_VVN earlier._NNU ,_, /div4._FU 2.5.5_MC Least_DAT squares_NN2 :_: weighted_JJ errors_NN2 In_II the_AT basic_JJ Gaussian_JJ method_NN1 just_RR described_VVN ,_, it_PPH1 has_VHZ been_VBN tacitly_RR assumed_VVN that_CST all_DB errors_NN2 are_VBR equally_RR likely_JJ ._. 
The_AT method_NN1 is_VBZ readily_RR adapted_VVN ,_, however_RR ,_, to_II cases_NN2 where_RRQ some_DD weighting_NN1 of_IO errors_NN2 is_VBZ desirable_JJ :_: it_PPH1 is_VBZ merely_RR a_AT1 case_NN1 of_IO premultiplying_VVG (_( 2.5.4.2_MC )_) by_II an_AT1 appropriate_JJ diagonal_JJ matrix_NN1 ._. 
Example_NN1 1_MC1 Suppose_VV0 we_PPIS2 wish_VV0 to_TO put_VVI a_AT1 straight_JJ line_NN1 &lsqb;_( formula_NN1 &rsqb;_) through_II the_AT following_JJ set_NN1 of_IO experimental_JJ points_NN2 :_: &lsqb;_( formula_NN1 &rsqb;_) Suppose_VV0 further_RRR that_DD1 ,_, in_II31 view_II32 of_II33 the_AT experimental_JJ method_NN1 ,_, we_PPIS2 may_VM expect_VVI errors_NN2 in_II f_ZZ1 to_TO be_VBI proportional_JJ to_II t_ZZ1 ._. 
Let_VV0 us_PPIO2 first_MD examine_VVI the_AT straight_JJ Gaussian_JJ solution_NN1 ._. 
The_AT equations_NN2 can_VM clearly_RR be_VBI written_VVN &lsqb;_( formula_NN1 &rsqb;_) Premultiplication_NN1 by_II AT_II yields_NN2 &lsqb;_( formula_NN1 &rsqb;_) so_CS21 that_CS22 the_AT optimum_JJ lime_NN1 ,_, if_CS all_DB errors_NN2 are_VBR equally_RR likely_JJ ,_, is_VBZ f_ZZ1 =_FO 0.57t_FO +_FO 1.80_MC ._. 
However_RR ,_, we_PPIS2 are_VBR given_CS21 that_CS22 errors_NN2 are_VBR likely_JJ to_TO be_VBI proportional_JJ to_II t_ZZ1 ._. 
To_TO make_VVI them_PPHO2 equally_RR likely_JJ ,_, we_PPIS2 must_VM multiply_VVI the_AT first_MD of_IO Equations_NN2 (_( 1_MC1 )_) ,_, at_II t_ZZ1 =_FO 1_MC1 ,_, by_II 6_MC ;_; the_AT second_NNT1 ,_, at_II t_ZZ1 =_FO 2_MC ,_, by_II 3_MC ;_; the_AT third_MD ,_, at_II t_ZZ1 =_FO 3_MC ,_, by_II 2_MC ,_, etc_RA ;_; i.e._REX we_PPIS2 premultiply_RR (_( 1_MC1 )_) by_II Diag(6,3,2,1,5,1,2,1)_FO ._. 
The_AT result_NN1 is_VBZ &lsqb;_( formula_NN1 &rsqb;_) and_CC premultiplication_NN1 of_IO this_DD1 by_II &lsqb;_( formula_NN1 &rsqb;_) yields_NN2 &lsqb;_( formula_NN1 &rsqb;_) to_II two_MC decimal_JJ places_NN2 ._. 
Hence_RR in_II this_DD1 case_NN1 ,_, the_AT optimum_JJ line_NN1 is_VBZ f_ZZ1 =_FO 0.50t_FO +_FO 2.01_MC ._. 
For_IF comparison_NN1 ,_, the_AT errors_NN2 e_ZZ1 in_II the_AT two_MC solutions_NN2 (_( 2_MC )_) and_CC (_( 3_MC )_) are_VBR (_( the_AT lines_NN2 intersect_VV0 at_II t_ZZ1 =3_FO )_) &lsqb;_( formula_NN1 &rsqb;_) and_CC it_PPH1 is_VBZ clear_JJ that_CST in_II the_AT second_MD case_NN1 the_AT errors_NN2 increase_VV0 markedly_RR with_IW t_ZZ1 ,_, as_CSA required_VVN ._. 
II_MC :_: EIGENVALUES_NN2 AND_CC VECTORS_NN2 2.6_MC THE_AT CHARACTERISTIC_JJ EQUATION_NN1 The_AT n_ZZ1 eigenvalues_NN2 of_IO a_AT1 square_JJ matrix_NN1 A_ZZ1 =_FO (_( Aij_NP1 )_) of_IO order_NN1 n_ZZ1 are_VBR given_VVN (_( see_VV0 1.16_MC )_) by_II the_AT roots_NN2 of_IO the_AT characteristic_JJ equation_NN1 &lsqb;_( formula_NN1 &rsqb;_) which_DDQ ,_, in_II expanded_JJ form_NN1 ,_, can_VM be_VBI written_VVN &lsqb;_( formula_NN1 &rsqb;_) In_II theory_NN1 ,_, therefore_RR ,_, the_AT calculation_NN1 of_IO the_AT eigenvalues_NN2 of_IO A_ZZ1 is_VBZ straightforward_JJ :_: we_PPIS2 form_VV0 the_AT characteristic_JJ determinant_NN1 in_II (_( 1_MC1 )_) ,_, expand_VV0 it_PPH1 to_II the_AT form_NN1 (_( 2_MC )_) ,_, and_CC solve_VV0 this_DD1 by_II any_DD suitable_JJ means_NN to_TO obtain_VVI &lsqb;_( formula_NN1 &rsqb;_) ._. 
Having_VHG obtained_VVN the_AT eigenvalues_NN2 ,_, can_VM readily_RR find_VVI the_AT corresponding_JJ eigenvectors_NN2 ._. 
If_CS &lsqb;_( formula_NN1 &rsqb;_) r_ZZ1 is_VBZ a_AT1 known_JJ eigenvalue_NN1 (_( which_DDQ we_PPIS2 assume_VV0 here_RL to_TO be_VBI unrepeated_JJ )_) and_CC &lsqb;_( formula_NN1 &rsqb;_) the_AT corresponding_JJ column_NN1 and_CC row_VV0 eigenvectors_NN2 respectively_RR ,_, then_RT &lsqb;_( formula_NN1 &rsqb;_) Since_CS &lsqb;_( formula_NN1 &rsqb;_) are_VBR each_DD1 arbitrary_JJ to_II a_AT1 scalar_JJ multiplier_NN1 ,_, we_PPIS2 can_VM in_II each_DD1 vector_NN1 make_VV0 a_AT1 suitable_JJ element_NN1 unity_NN1 ;_; then_RT Equations_NN2 (_( 3_MC )_) provide_VV0 in_II each_DD1 case_NN1 n_ZZ1 linear_JJ algebraic_JJ equations_NN2 for_IF the_AT (_( n_ZZ1 1_MC1 )_) unknown_JJ elements_NN2 :_: one_MC1 equation_NN1 is_VBZ superfluous_JJ ,_, or_CC can_VM be_VBI used_VVN as_II a_AT1 check_NN1 ._. 
For_REX21 example_REX22 ,_, if_CS we_PPIS2 make_VV0 the_AT last_MD element_NN1 of_IO &lsqb;_( formula_NN1 &rsqb;_) unity_NN1 ,_, and_CC partition_NN1 (_( &lsqb;_( formula_NN1 &rsqb;_) )_) to_TO isolate_VVI its_APPGE last_MD column_NN1 and_CC row_NN1 ,_, we_PPIS2 can_VM write_VVI (_( 3_MC )_) as_CSA &lsqb;_( formula_NN1 &rsqb;_) which_DDQ give_VV0 &lsqb;_( formula_NN1 &rsqb;_) Equations_NN2 (_( 5_MC )_) can_VM be_VBI solved_VVN by_II any_DD of_IO the_AT methods_NN2 of_IO 2.5_MC (_( the_AT second_NNT1 would_VM be_VBI transposed_VVN to_II &lsqb;_( formula_NN1 &rsqb;_) ,_, and_CC then_RT &lsqb;_( formula_NN1 &rsqb;_) Unfortunately_RR ,_, so_RG far_RR as_CSA the_AT expansion_NN1 of_IO (_( 1_MC1 )_) to_II (_( 2_MC )_) is_VBZ concerned_JJ ,_, theory_NN1 and_CC practice_NN1 are_VBR at_II odds_NN2 ._. 
With_IW &lsqb;_( formula_NN1 &rsqb;_) general_NN1 ,_, the_AT labour_NN1 involved_JJ in_II the_AT expansion_NN1 is_VBZ quite_RG prohibitive_JJ even_RR for_IF small_JJ values_NN2 of_IO n_ZZ1 ,_, and_CC indirect_JJ methods_NN2 are_VBR always_RR used_VVN in_II practice_NN1 ._. 
We_PPIS2 describe_VV0 two_MC such_DA methods_NN2 below_RL ;_; first_MD ,_, however_RR ,_, we_PPIS2 observe_VV0 that_CST ,_, in_II (_( 2_MC )_) ,_, we_PPIS2 can_VM readily_RR obtain_VVI the_AT two_MC coefficients_NN2 p1_FO ,_, pn_NNU ,_, since_CS (_( see_VV0 1.22_MC ,_, Theorem_NN1 II_MC ,_, III_MC )_) &lsqb;_( formula_NN1 &rsqb;_) The_AT evaluation_NN1 of_IO a_AT1 numerical_JJ determinant_NN1 ,_, even_RR of_IO high_JJ order_NN1 ,_, is_VBZ very_RG quickly_RR accomplished_VVN on_II any_DD modern_JJ computer_NN1 by_II reduction_NN1 to_II triangular_JJ form_NN1 ;_; thus_RR we_PPIS2 can_VM find_VVI p1_FO and_CC pn_NNU without_IW difficulty_NN1 ._. 
It_PPH1 remains_VVZ ,_, however_RR ,_, to_TO find_VVI p2_FO ,_, p3_FO ,_, ..._... ,_, pn-1._MC 2.6.1_MC Location_NN1 method_NN1 In_II Equation_NN1 (_( 2.6.1_MC )_) we_PPIS2 arbitrarily_RR assign_VV0 (_( n_ZZ1 2_MC )_) different_JJ values_NN2 to_II &lsqb;_( formula_NN1 &rsqb;_) (_( other_II21 than_II22 &lsqb;_( formula_NN1 &rsqb;_) =_FO 0_MC ,_, which_DDQ we_PPIS2 have_VH0 in_II effect_NN1 used_VMK to_TO determine_VVI pn_NNU )_) and_CC then_RT evaluate_VV0 the_AT (_( numerical_JJ )_) determinant_NN1 for_IF each_DD1 of_IO these_DD2 ._. 
We_PPIS2 then_RT obtain_VV0 n_ZZ1 2_MC linear_JJ algebraic_JJ equations_NN2 for_IF the_AT n_ZZ1 2_MC unknowns_NN2 pi_NN1 ._. 
A_AT1 caution_NN1 should_VM be_VBI added_VVN :_: most_DAT the_AT arbitrary_JJ values_NN2 of_IO &lsqb;_( formula_NN1 &rsqb;_) should_VM preferably_RR lie_VVI among_II the_AT zeros_NN2 of_IO &lsqb;_( formula_NN1 &rsqb;_) (_( &lsqb;_( formula_NN1 &rsqb;_) ;_; otherwise_RR inordinately_RR high_JJ accuracy_NN1 may_VM be_VBI needed_VVN ._. 
At_II this_DD1 stage_NN1 ,_, the_AT zeros_NN2 of_IO &lsqb;_( formula_NN1 &rsqb;_) (_( &lsqb;_( formula_NN1 &rsqb;_) )_) are_VBR now_RT known_VVN ,_, but_CCB their_APPGE approximate_JJ locations_NN2 may_VM often_RR be_VBI inferred_VVN from_II their_APPGE sum_NN1 and_CC product_NN1 as_CSA given_VVN by_II (_( 2.6.6_MC )_) and_CC (_( 2.6.7_MC )_) ._. 
Example_NN1 1_MC1 Let_VV0 &lsqb;_( formula_NN1 &rsqb;_) The_AT sum_NN1 of_IO the_AT eigenvalues_NN2 is_VBZ thus_RR &lsqb;_( formula_NN1 &rsqb;_) and_CC their_APPGE product_NN1 3906.25_MC (_( =p4_FO )_) ._. 
Their_APPGE arithmetic_NN1 mean_NN1 is_VBZ thus_RR about_RG 11_MC an_AT1 the_AT fourth_MD root_NN1 of_IO &lsqb;_( formula_NN1 &rsqb;_) about_RG 8_MC ._. 
These_DD2 suggest_VV0 that_CST we_PPIS2 might_VM use_VVI &lsqb;_( formula_NN1 &rsqb;_) =_FO 20_MC ,_, &lsqb;_( formula_NN1 &rsqb;_) =_FO 10_MC in_II (_( 2.6.1_MC )_) ;_; we_PPIS2 have_VH0 already_RR used_VVN &lsqb;_( formula_NN1 &rsqb;_) =_FO 0_MC to_TO obtain_VVI &lsqb;_( formula_NN1 &rsqb;_) ._. 
It_PPH1 is_VBZ readily_RR found_VVN ,_, by_II condensing_VVG the_AT numerical_JJ determinants_NN2 ,_, &lsqb;_( formula_NN1 &rsqb;_) Now_RT we_PPIS2 know_VV0 that_CST &lsqb;_( formula_NN1 &rsqb;_) or_CC &lsqb;_( formula_NN1 &rsqb;_) Insertion_NN1 of_IO &lsqb;_( formula_NN1 &rsqb;_) =_FO 20_MC ,_, &lsqb;_( formula_NN1 &rsqb;_) =_FO 10_MC in_II (_( 3_MC )_) together_RL with_IW the_AT numerical_JJ values_NN2 of_IO &lsqb;_( formula_NN1 &rsqb;_) (_( gl_NNU )_) from_II (_( 2_MC )_) yields_VVZ the_AT two_MC equations_NN2 &lsqb;_( formula_NN1 &rsqb;_) which_DDQ give_VV0 p2_FO =_FO 606.25_MC ,_, p3_FO =_FO 2812.5_MC ._. 
Hence_RR &lsqb;_( formula_NN1 &rsqb;_) on_II factorisation_NN1 ._. 
The_AT eigenvalues_NN2 of_IO A_ZZ1 are_VBR thus_RR 25_MC ,_, 12.5_MC ,_, 5_MC ,_, 2.5_MC ._. 
We_PPIS2 illustrate_VV0 the_AT calculation_NN1 of_IO the_AT eigenvector_NN1 &lsqb;_( formula_NN1 &rsqb;_) corresponding_VVG to_II the_AT eigenvalue_NN1 12.5_MC ._. 
Then_RT &lsqb;_( formula_NN1 &rsqb;_) giving_VVG &lsqb;_( formula_NN1 &rsqb;_) of_IO which_DDQ the_AT solution_NN1 is_VBZ (_( any_DD method_NN1 from_II 2.5_MC may_VM be_VBI used_VVN )_) &lsqb;_( formula_NN1 &rsqb;_) and_CC it_PPH1 is_VBZ readily_RR checked_VVN that_CST these_DD2 also_RR satisfy_VV0 the_AT fourth_MD equation._NNU 2.6.2_MC Iteration_NN1 method_NN1 In_II31 view_II32 of_II33 the_AT Cayley-Hamilton_NP1 theorem_NN1 ,_, A_ZZ1 satisfies_VVZ its_APPGE own_DA characteristic_JJ equation_NN1 ;_; i.e._REX &lsqb;_( formula_NN1 &rsqb;_) Postmultiply_RR this_DD1 by_II any_DD arbitrary_JJ column_NN1 co_NN1 and_CC write_VV0 &lsqb;_( formula_NN1 &rsqb;_) ._. 
Then_RT a_AT1 computer_NN1 will_VM rapidly_RR evaluate_VVI the_AT successive_JJ products_NN2 &lsqb;_( formula_NN1 &rsqb;_) and_RR31 so_RR32 on_RR33 ._. 
Accordingly_RR (_( 1_MC1 )_) yields_NN2 &lsqb;_( formula_NN1 &rsqb;_) This_DD1 provided_CS n_ZZ1 linear_JJ algebraic_JJ equations_NN2 for_IF the_AT n_ZZ1 unknowns_NN2 pi_NN1 ._. 
As_CSA before_RT ,_, we_PPIS2 can_VM find_VVI &lsqb;_( formula_NN1 &rsqb;_) independently_RR if_CS we_PPIS2 so_RR wish_VV0 ._. 
It_PPH1 must_VM be_VBI added_VVN ,_, however_RR ,_, that_CST this_DD1 method_NN1 is_VBZ not_XX suitable_JJ when_CS n_ZZ1 is_VBZ large_JJ ._. 
As_CSA we_PPIS2 shall_VM see_VVI later_RRR ,_, the_AT columns_NN2 ci_MC usually_RR tend_VV0 to_II proportionality_NN1 as_CSA i_ZZ1 increases_VVZ ,_, so_CS21 that_CS22 Equations_NN2 (_( 2_MC )_) become_VV0 increasingly_RR ill-conditioned_JJ ;_; also_RR the_AT numbers_NN2 involved_VVN can_VM assume_VVI widely_RR different_JJ magnitudes_NN2 ._. 
Example_NN1 1_MC1 We_PPIS2 use_VV0 the_AT matrix_NN1 A_ZZ1 of_IO (_( 2.6.1.1_MC )_) with_IW co_NN1 =_FO e4_FO ._. 
It_PPH1 is_VBZ found_VVN that_CST successive_JJ columns_NN2 ci_MC are_VBR then_RT &lsqb;_( formula_NN1 &rsqb;_) Accordingly_RR ,_, (_( 2_MC )_) can_VM be_VBI written_VVN &lsqb;_( formula_NN1 &rsqb;_) in_II agreement_NN1 with_IW (_( 2.6.1.4_MC )_) ._. 
With_IW the_AT particular_JJ choice_NN1 of_IO co_NN1 ,_, p4_FO enters_VVZ into_II the_AT last_MD equation_NN1 only_RR ._. 
Accordingly_RR ,_, if_CS we_PPIS2 calculate_VV0 p1_FO ,_, p4_FO from_II the_AT trace_NN1 and_CC determinant_NN1 of_IO A_ZZ1 ,_, we_PPIS2 can_VM discard_VVI the_AT last_MD equation_NN1 and_CC substitute_VVI for_IF p1_FO in_II the_AT remainder_NN1 ,_, when_RRQ (_( 3_MC )_) reduces_VVZ to_II &lsqb;_( formula_NN1 &rsqb;_) Any_DD two_MC of_IO these_DD2 (_( consistent_JJ )_) equations_NN2 give_VV0 p2_FO ,_, p3_FO their_APPGE values_NN2 in_II (_( 4_MC )_) ._. 
2.7_MC POWER_NN1 METHODS_NN2 The_AT most_RGT commonly_RR used_JJ device_NN1 for_IF finding_VVG the_AT eigenvalues_NN2 and_CC vectors_NN2 of_IO a_AT1 matrix_NN1 A_ZZ1 is_VBZ the_AT power_NN1 method_NN1 ,_, or_CC one_MC1 of_IO its_APPGE many_DA2 variants_NN2 ,_, in_II which_DDQ A_ZZ1 is_VBZ effectively_RR raised_VVN to_II a_AT1 high_JJ power_NN1 ._. 
We_PPIS2 illustrate_VV0 some_DD aspects_NN2 of_IO the_AT basic_JJ principle._NNU 2.7.1_MC A_ZZ1 dominant_JJ eigenvalue_NN1 Sylvester_NP1 's_GE expansion_NN1 for_IF A_ZZ1 is_VBZ (_( see_VV0 1.18_MC )_) ,_, in_II its_APPGE simplest_JJT form_NN1 ,_, &lsqb;_( formula_NN1 &rsqb;_) and_CC in_II31 view_II32 of_II33 the_AT properties_NN2 of_IO the_AT constituent_NN1 matrices_NN2 &lsqb;_( formula_NN1 &rsqb;_) ,_, &lsqb;_( formula_NN1 &rsqb;_) Since_CS the_AT order_NN1 of_IO the_AT terms_NN2 on_II the_AT right_NN1 is_VBZ immaterial_JJ ,_, there_EX is_VBZ no_AT loss_NN1 of_IO generality_NN1 i_ZZ1 to_TO be_VBI the_AT eigenvalue_NN1 of_IO greatest_JJT modulus_NN1 ._. 
If_CS for_IF the_AT moment_NN1 we_PPIS2 assume_VV0 it_PPH1 to_TO be_VBI real_JJ and_CC unrepeated_JJ ,_, then_RT all_DB other_JJ eigenvalues_NN2 have_VH0 smaller_JJR moduli_NN2 ._. 
Thus_RR if_CS r_ZZ1 is_VBZ increased_VVN sufficiently_RR ,_, say_VV0 to_II a_AT1 value_NN1 s_ZZ1 ,_, the_AT first_MD term_NN1 on_II the_AT right_NN1 of_IO (_( 2_MC )_) dominates_VVZ the_AT rest_NN1 ,_, which_DDQ become_VV0 relatively_RR insignificant_JJ ._. 
Then_RT ,_, in_II the_AT limit_NN1 ,_, &lsqb;_( formula_NN1 &rsqb;_) By_II use_NN1 of_IO (_( 3_MC )_) we_PPIS2 thus_RR find_VV0 both_RR &lsqb;_( formula_NN1 &rsqb;_) 1_MC1 and_CC &lsqb;_( formula_NN1 &rsqb;_) ._. 
We_PPIS2 could_VM in_II theory_NN1 find_VV0 them_PPHO2 from_II As_RG only_RR ,_, since_CS &lsqb;_( formula_NN1 &rsqb;_) ,_, but_CCB there_EX is_VBZ a_AT1 practical_JJ difficulty_NN1 ._. 
Unless_CS the_AT modulus_NN1 of_IO &lsqb;_( formula_NN1 &rsqb;_) 1_MC1 is_VBZ close_JJ to_II unity_NN1 ,_, the_AT numbers_NN2 appearing_VVG in_II Ar_UH (_( r_ZZ1 large_JJ )_) are_VBR either_RR inordinately_RR large_JJ or_CC small_JJ ;_; in_II practice_NN1 therefore_RR it_PPH1 is_VBZ usual_JJ to_TO find_VVI not_XX Ar_UH but_CCB a_AT1 matrix_NN1 proportional_JJ to_II it_PPH1 :_: at_II each_DD1 stage_NN1 in_II the_AT power_NN1 evaluation_NN1 ,_, a_AT1 homologous_JJ element_NN1 is_VBZ reduced_VVN to_II unity_NN1 ._. 
Example_NN1 1_MC1 Let_VV0 A_ZZ1 be_VBI the_AT matrix_NN1 in_II Equation_NN1 (_( 2.6.1.1_MC )_) ._. 
Then_RT ,_, if_CS we_PPIS2 reduce_VV0 the_AT bottom_JJ right-hand_JJ element_NN1 to_II unity_NN1 ,_, we_PPIS2 have_VH0 &lsqb;_( formula_NN1 &rsqb;_) We_PPIS2 now_RT evaluate_VV0 B2_FO and_CC divide_VV0 through_RP by_II the_AT bottom_JJ right-hand_JJ element_NN1 to_TO obtain_VVI &lsqb;_( formula_NN1 &rsqb;_) Evaluation_NN1 of_IO F2/2.4997_FU repeats_VVZ F_ZZ1 except_II21 for_II22 occasional_JJ small_JJ differences_NN2 in_II the_AT fourth_MD decimal_JJ place_NN1 ._. 
Now_RT ,_, if_CS we_PPIS2 round_VV0 off_RP to_II three_MC decimal_JJ places_NN2 ,_, we_PPIS2 find_VV0 that_CST F_ZZ1 can_VM be_VBI written_VVN as_II the_AT unit_NN1 rank_NN1 matrix_NN1 ,_, proportional_JJ to_II &lsqb;_( formula_NN1 &rsqb;_) &lsqb;_( formula_NN1 &rsqb;_) If_CS we_PPIS2 now_RT revert_VV0 to_II A_ZZ1 and_CC postmultiply_RR by_II &lsqb;_( formula_NN1 &rsqb;_) &lsqb;_( formula_NN1 &rsqb;_) Thus_RR ,_, &lsqb;_( formula_NN1 &rsqb;_) 1_MC1 =_FO 25_MC ._. 
Also_RR ,_, the_AT inner_JJ product_NN1 &lsqb;_( formula_NN1 &rsqb;_) is_VBZ &lsqb;_( formula_NN1 &rsqb;_) so_CS that_DD1 normalising_VVG to_TO make_VVI &lsqb;_( formula_NN1 &rsqb;_) unity_NN1 :_: Note_VV0 that_CST &lsqb;_( formula_NN1 &rsqb;_) so_CS21 that_CS22 A_ZZ1 has_VHZ been_VBN raised_VVN fairly_RR rapidly_RR to_II a_AT1 high_JJ power_NN1 ._. 
An_AT1 alternative_NN1 ._. 
There_EX is_VBZ ,_, however_RR ,_, an_AT1 alternative_JJ device_NN1 which_DDQ is_VBZ in_II practice_NN1 more_RGR commonly_RR used_VVN ;_; this_DD1 is_VBZ repeated_VVN premultiplication_NN1 by_II A_ZZ1 of_IO an_AT1 arbitrary_JJ column_NN1 &lsqb;_( formula_NN1 &rsqb;_) as_CSA in_II 2.6.2_MC ._. 
We_PPIS2 successively_RR evaluate_VV0 &lsqb;_( formula_NN1 &rsqb;_) etc_RA ._. 
If_CS we_PPIS2 postmultiply_RR (_( 2_MC )_) by_II co_NN1 and_CC write_VV0 &lsqb;_( formula_NN1 &rsqb;_) (_( ki_NN2 is_VBZ of_RR21 course_RR22 a_AT1 scalar_JJ )_) we_PPIS2 obtain_VV0 &lsqb;_( formula_NN1 &rsqb;_) Accordingly_RR ,_, provided_CS21 that_CS22 co_NN1 is_VBZ not_XX such_CS21 that_CS22 k1_FO (_( &lsqb;_( formula_NN1 &rsqb;_) )_) vanishes_VVZ ,_, the_AT first_MD term_NN1 on_II the_AT right_NN1 dominates_VVZ the_AT rest_NN1 as_CSA r_ZZ1 increases_VVZ ,_, until_CS in_II the_AT limit_NN1 when_CS r_ZZ1 =_FO s_ZZ1 &lsqb;_( formula_NN1 &rsqb;_) Example_NN1 2_MC We_PPIS2 exemplify_VV0 this_DD1 using_VVG the_AT same_DA matrix_NN1 A_ZZ1 as_CSA above_RL ;_; also_RR ,_, we_PPIS2 adopt_VV0 the_AT device_NN1 of_IO reducing_VVG at_II each_DD1 stage_NN1 a_AT1 homologous_JJ element_NN1 (_( in_II this_DD1 case_NN1 the_AT bottom_JJ element_NN1 )_) to_II unity_NN1 ,_, beginning_VVG with_IW co_NN1 =_FO e4_FO ._. 
The_AT first_MD step_NN1 is_VBZ &lsqb;_( formula_NN1 &rsqb;_) Proceeding_VVG in_II this_DD1 way_NN1 ,_, we_PPIS2 find_VV0 successive_JJ columns_NN2 as_CSA in_II Table_NN1 1_MC1 ._. 
Also_RR given_VVN are_VBR the_AT dividing_JJ factors_NN2 at_II each_DD1 step_NN1 ._. 
Since_CS c16_FO repeats_VVZ c15_FO ,_, they_PPHS2 give_VV0 x1_FO ;_; also_RR the_AT dividing_JJ factor_NN1 25_MC is_VBZ &lsqb;_( formula_NN1 &rsqb;_) 1_MC1 ._. 
To_TO complete_VVI the_AT solution_NN1 ,_, we_PPIS2 now_RT require_VV0 to_TO find_VVI &lsqb;_( formula_NN1 &rsqb;_) ;_; this_DD1 could_VM be_VBI done_VDN by_II repeating_VVG the_AT iterative_JJ procedure_NN1 ,_, evaluating_VVG &lsqb;_( formula_NN1 &rsqb;_) ,_, where_RRQ &lsqb;_( formula_NN1 &rsqb;_) is_VBZ an_AT1 arbitrary_JJ row_NN1 ,_, or_CC more_RGR usually_RR ,_, by_II the_AT device_NN1 given_VVN in_II (_( 2.6.4_MC )_) ,_, viz_REX :_: &lsqb;_( formula_NN1 &rsqb;_) giving_VVG &lsqb;_( formula_NN1 &rsqb;_) We_PPIS2 leave_VV0 it_PPH1 to_II the_AT reader_NN1 to_TO show_VVI that_CST in_II this_DD1 example_NN1 ,_, (_( 7_MC )_) leads_VVZ to_II &lsqb;_( formula_NN1 &rsqb;_) or_CC to_II a_AT1 scalar_JJ multiple_NN1 of_IO it_PPH1 ._. 
We_PPIS2 normalise_VV0 &lsqb;_( formula_NN1 &rsqb;_) in_II the_AT usual_JJ way_NN1 ._. 
We_PPIS2 may_VM note_VVI in_II passing_VVG that_CST (_( as_CSA they_PPHS2 should_VM )_) the_AT columns_NN2 &lsqb;_( formula_NN1 &rsqb;_) and_CC &lsqb;_( formula_NN1 &rsqb;_) of_IO Table_NN1 1_MC1 agree_VV0 with_IW the_AT last_MD columns_NN2 of_IO B_ZZ1 ,_, C_ZZ1 ,_, D_ZZ1 ,_, E_ZZ1 ,_, F_ZZ1 (_( subject_II21 to_II22 rounding-off_JJ errors_NN2 in_II the_AT last_MD place_NN1 of_IO decimals_NN2 )_) ._. 
This_DD1 leads_VVZ naturally_RR to_II a_AT1 comparison_NN1 of_IO the_AT two_MC methods_NN2 ._. 
There_EX are_VBR two_MC reasons_NN2 why_RRQ the_AT second_MD method_NN1 is_VBZ commonly_RR used_VVN ._. 
First_MD ,_, if_CS an_AT1 error_NN1 (_( even_RR of_IO rounding-off_JJ only_JJ )_) is_VBZ made_VVN during_II the_AT successive_JJ squaring_JJ process_NN1 ,_, it_PPH1 is_VBZ multiplied_VVN by_II itself_PPX1 &lsqb;_( formula_NN1 &rsqb;_) subsequently_RR ,_, an_AT1 persists_VVZ through_II the_AT calculations_NN2 ;_; in_II the_AT second_MD method_NN1 ,_, if_CS an_AT1 error_NN1 is_VBZ made_VVN in_II calculating_VVG ci_MC ,_, this_DD1 only_JJ amounts_NN2 to_II choosing_VVG a_AT1 new_JJ arbitrary_JJ column_NN1 ,_, an_AT1 at_RR21 worst_RR22 may_VM prolong_VVI the_AT calculations_NN2 ._. 
But_CCB ,_, mainly_RR ,_, the_AT first_MD method_NN1 usually_RR involves_VVZ more_DAR work_NN1 than_CSN the_AT second_NNT1 ._. 
It_PPH1 is_VBZ difficult_JJ to_TO quantify_VVI &quot;_" work_NN1 &quot;_" exactly_RR ,_, but_CCB if_CS the_AT number_NN1 of_IO individual_JJ multiplications_NN2 is_VBZ taken_VVN as_II a_AT1 guide_NN1 ,_, then_RT ,_, if_CS A_ZZ1 is_VBZ of_IO order_NN1 n_ZZ1 ,_, each_DD1 of_IO the_AT n2_FO elements_NN2 of_IO &lsqb;_( formula_NN1 &rsqb;_) involves_VVZ n_ZZ1 multiplications_NN2 :_: i.e._REX n3_FO in_II all_DB ._. 
If_CS r_ZZ1 squaring_JJ processes_NN2 are_VBR needed_VVN ,_, we_PPIS2 perform_VV0 rn3_FO multiplications_NN2 ,_, yielding_VVG &lsqb;_( formula_NN1 &rsqb;_) ,_, r_ZZ1 must_VM be_VBI an_AT1 integer_NN1 ._. 
In_II the_AT column_NN1 process_NN1 ,_, each_DD1 step_NN1 involves_VVZ n2_FO multiplications_NN2 ;_; hence_RR ,_, to_TO reach_VVI the_AT same_DA accuracy_NN1 ,_, between_II &lsqb;_( formula_NN1 &rsqb;_) and_CC &lsqb;_( formula_NN1 &rsqb;_) multiplications_NN2 are_VBR needed_VVN ._. 
Now_RT ,_, provided_CS the_AT eigenvalues_NN2 are_VBR reasonably_RR separated_VVN ,_, column_NN1 iterations_NN2 usually_RR number_NN1 between_II 15_MC and_CC 50_MC ._. 
Suppose_VV0 ,_, in_II a_AT1 particular_JJ case_NN1 ,_, 25_MC are_VBR required_VVN ._. 
Then_RT the_AT &quot;_" work_NN1 &quot;_" is_VBZ measured_VVN as_CSA &lsqb;_( formula_NN1 &rsqb;_) From_II Table_NN1 2_MC ,_, r_ZZ1 must_VM be_VBI 5_MC ,_, with_IW the_AT work_NN1 measure_NN1 for_IF squaring_VVG 5n3_FO ._. 
The_AT ratio_NN1 is_VBZ n/5_FU ;_; so_RR if_CS &lsqb;_( formula_NN1 &rsqb;_) the_AT work_NN1 involved_JJ in_II squaring_VVG is_VBZ greater_JJR ._. 
If_CS ,_, say_VV0 ,_, n_ZZ1 is_VBZ 100_MC ,_, the_AT work_NN1 is_VBZ 20_MC times_NNT2 greater._NNU &lsqb;_( formula_NN1 &rsqb;_) 2.7.2_MC Subdominant_JJ eigenvalues_NN2 :_: deflation_NN1 Again_RT ,_, for_IF the_AT moment_NN1 we_PPIS2 assume_VV0 the_AT eigenvalues_NN2 of_IO A_ZZ1 to_TO be_VBI real_JJ and_CC unrepeated_JJ ;_; and_CC we_PPIS2 assume_VV0 that_CST in_II (_( 2.7.1.1_MC )_) they_PPHS2 appear_VV0 in_II descending_JJ order_NN1 of_IO modulus_NN1 ._. 
If_CS ,_, as_CSA in_II 2.7.1_MC we_PPIS2 have_VH0 found_VVN &lsqb;_( formula_NN1 &rsqb;_) ,_, then_RT Equation_NN1 (_( 2.7.1.1_MC )_) can_VM be_VBI written_VVN &lsqb;_( formula_NN1 &rsqb;_) The_AT known_JJ matrix_NN1 on_II the_AT left_JJ now_RT has_VHZ all_DB the_AT eigenvalues_NN2 and_CC vectors_NN2 of_IO A_ZZ1 except_II the_AT first_MD set_NN1 ,_, and_CC it_PPH1 can_VM be_VBI treated_VVN as_CSA in_II 2.7.1_MC ._. 
The_AT process_NN1 of_IO reducing_VVG A_ZZ1 in_II this_DD1 way_NN1 is_VBZ known_VVN as_II &quot;_" deflation_NN1 &quot;_" ._. 
Evidently_RR the_AT eigenvalue_NN1 &lsqb;_( formula_NN1 &rsqb;_) 1_MC1 has_VHZ been_VBN replaced_VVN by_II zero_MC ,_, so_CS21 that_CS22 A1_FO is_VBZ singular_JJ ._. 
Having_VHG evaluated_VVN A1_FO ,_, we_PPIS2 can_VM again_RT apply_VVI column_NN1 iteration_NN1 ._. 
However_RR ,_, we_PPIS2 need_VM not_XX begin_VVI with_IW an_AT1 arbitrary_JJ column_NN1 ;_; we_PPIS2 can_VM readily_RR find_VVI an_AT1 appropriate_JJ column_NN1 to_TO begin_VVI with_IW ,_, which_DDQ avoids_VVZ many_DA2 iteration_NN1 steps_NN2 ._. 
An_AT1 example_NN1 will_VM make_VVI this_DD1 clear_JJ ._. 
Example_NN1 1_MC1 We_PPIS2 continue_VV0 Example_NN1 2.7.1.1_MC ._. 
The_AT iterations_NN2 have_VH0 shown_VVN that_CST &lsqb;_( formula_NN1 &rsqb;_) 1_MC1 =_FO 25_MC ,_, while_CS x1_FO is_VBZ proportional_JJ to_II 1,2,3,5_MC and_CC &lsqb;_( formula_NN1 &rsqb;_) to_II (_( 1,7,5,4_MC )_) ._. 
Normalising_VVG these_DD2 ,_, we_PPIS2 find_VV0 &lsqb;_( formula_NN1 &rsqb;_) Note_VV0 that_CST &lsqb;_( formula_NN1 &rsqb;_) vanishes_VVZ when_CS premultiplied_VVN by_II &lsqb;_( formula_NN1 &rsqb;_) ._. 
We_PPIS2 now_RT revert_VV0 to_II the_AT iterations_NN2 for_IF &lsqb;_( formula_NN1 &rsqb;_) 1_MC1 an_AT1 observe_VV0 that_CST ,_, for_REX21 example_REX22 ,_, c9_FO of_IO Table_NN1 2.7.1.1_MC is_VBZ mostly_RR composed_VVN of_IO x1_FO ._. 
The_AT remainder_NN1 will_VM clearly_RR be_VBI mostly_RR &lsqb;_( formula_NN1 &rsqb;_) ._. 
Since_CS x1_FO is_VBZ orthogonal_JJ to_II &lsqb;_( formula_NN1 &rsqb;_) ,_, the_AT part_NN1 of_IO c9_FO due_II21 to_II22 x1_FO will_VM disappear_VVI when_RRQ it_PPH1 is_VBZ premultiplied_VVN by_II A1_FO (_( see_VV0 1_MC1 )_) )_) )_) ._. 
Thus_RR we_PPIS2 find_VV0 &lsqb;_( formula_NN1 &rsqb;_) and_CC dividing_VVG by_II the_AT last_MD element_NN1 ,_, we_PPIS2 find_VV0 as_II a_AT1 starting_NN1 column_NN1 co_NN1 for_IF iteration_NN1 with_IW A1_FO &lsqb;_( formula_NN1 &rsqb;_) The_AT iterations_NN2 give_VV0 us_PPIO2 successive_JJ columns_NN2 as_CSA in_II Table_NN1 1_MC1 ._. 
Column_NN1 c9_FO repeats_VVZ c8_FO ,_, so_CS21 that_CS22 &lsqb;_( formula_NN1 &rsqb;_) 2_MC =_FO 12.5_MC and_CC &lsqb;_( formula_NN1 &rsqb;_) is_VBZ proportional_JJ to_II -0.5_MC ,_, =_FO -0.5,0,1_MC ._. 
As_CSA before_RT ,_, we_PPIS2 now_RT find_VV0 that_CST &lsqb;_( formula_NN1 &rsqb;_) is_VBZ proportional_JJ to_II (_( -1_MC ,_, -2,0,1_MC )_) ._. 
Normalising_NP1 ,_, we_PPIS2 find_VV0 &lsqb;_( formula_NN1 &rsqb;_) We_PPIS2 proceed_VV0 to_II the_AT third_MD eigenvalue_NN1 ;_; we_PPIS2 further_RRR deflate_VV0 A_ZZ1 by_II subtracting_VVG the_AT unit_NN1 rank_NN1 matrix_NN1 (_( 3_MC )_) from_II A1_FO :_: &lsqb;_( formula_NN1 &rsqb;_) Note_VV0 that_CST &lsqb;_( formula_NN1 &rsqb;_) =_FO 7.5_MC ,_, while_CS &lsqb;_( formula_NN1 &rsqb;_) is_VBZ in_II fact_NN1 doubly_RR degenerate_JJ ;_; it_PPH1 vanishes_VVZ when_CS premultiplied_VVN by_II &lsqb;_( formula_NN1 &rsqb;_) or_CC &lsqb;_( formula_NN1 &rsqb;_) ._. 
As_CSA before_RT ,_, we_PPIS2 choose_VV0 a_AT1 column_NN1 from_II Table_NN1 1_MC1 say_VV0 c3_FO =_FO -0.4978_MC ,_, -0.4992_MC ,_, -0.0015,1_MC and_CC apply_VV0 it_PPH1 as_II a_AT1 postmultiplier_JJR to_II &lsqb;_( formula_NN1 &rsqb;_) ._. 
The_AT result_NN1 is_VBZ a_AT1 column_NN1 proportional_JJ to_II 0.981_MC ,_, 0.0095_MC ,_, 1.0095_MC ,_, 1_MC1 ._. 
We_PPIS2 leave_VV0 it_PPH1 to_II the_AT reader_NN1 to_TO show_VVI that_CST this_DD1 leads_VVZ rapidly_RR to_II &lsqb;_( formula_NN1 &rsqb;_) 3_MC =_FO 5_MC with_IW x3_FO proportional_JJ to_II 1,0_MC ,_, 1,1_MC an_AT1 that_DD1 &lsqb;_( formula_NN1 &rsqb;_) is_VBZ then_RT found_VVN to_TO be_VBI proportional_JJ to_II (_( 27_MC ,_, -11_MC ,_, -15.8_MC )_) ._. 
We_PPIS2 normalise_VV0 this_DD1 to_TO obtain_VVI &lsqb;_( formula_NN1 &rsqb;_) an_AT1 now_RT further_RRR deflate_VV0 A_ZZ1 to_TO obtain_VVI A3_FO ;_; in_II this_DD1 case_NN1 ,_, as_CSA will_VM be_VBI seen_VVN ,_, no_AT iteration_NN1 is_VBZ required_VVN to_TO obtain_VVI a_AT1 solution_NN1 :_: &lsqb;_( formula_NN1 &rsqb;_) In_II obtaining_VVG the_AT last_MD form_NN1 of(5)_FO we_PPIS2 have_VH0 noted_VVN (_( a_ZZ1 )_) that_CST &lsqb;_( formula_NN1 &rsqb;_) ;_; since_CS the_AT other_JJ three_MC eigenvalues_NN2 have_VH0 been_VBN replaced_VVN by_II zeros_NN2 ,_, &lsqb;_( formula_NN1 &rsqb;_) 4_MC =_FO 2.5_MC ;_; (_( b_ZZ1 )_) A3_FO is_VBZ ,_, by_II inspection_NN1 ,_, of_IO unit_NN1 rank_NN1 an_AT1 so_RR ,_, with_IW &lsqb;_( formula_NN1 &rsqb;_) 4_MC extracted_VVD ,_, can_VM be_VBI written_VVN in_II the_AT form_NN1 (_( 5_MC )_) ._. 
This_DD1 completes_VVZ the_AT solution_NN1 ._. 
With_IW the_AT four_MC terms_NN2 of_IO the_AT Sylvester_NP1 expansion_NN1 now_RT given_VVN numerically_RR in_II (_( 2_MC )_) ,_, (_( 3_MC )_) ,_, (_( 4_MC )_) and_CC (_( 5_MC )_) ,_, the_AT whole_JJ solution_NN1 can_VM be_VBI expressed_VVN compendiously_RR as_CSA &lsqb;_( formula_NN1 &rsqb;_) 2.7.3_MC Subdominant_JJ eigenvalues_NN2 :_: sweeping_JJ Although_CS the_AT delation_NN1 procedure_NN1 discussed_VVN in_II 2.7.2_MC is_VBZ the_AT most_RGT straightforward_JJ possible_JJ method_NN1 ,_, other_JJ devices_NN2 are_VBR often_RR useful_JJ and_CC sometimes_RT preferable_JJ ._. 
We_PPIS2 now_RT discuss_VV0 &quot;_" sweeping_JJ &quot;_" ,_, which_DDQ takes_VVZ two_MC forms._NNU (_( i_ZZ1 )_) First_MD method_NN1 Instead_II21 of_II22 operating_VVG with_IW a_AT1 deflated_JJ matrix_NN1 ,_, we_PPIS2 operate_VV0 on_II the_AT original_JJ matrix_NN1 A_ZZ1 (_( or_CC a_AT1 part_NN1 of_IO it_PPH1 )_) with_IW a_AT1 column_NN1 vector_NN1 from_II which_DDQ all_DB contributions_NN2 from_II x1_FO have_VH0 been_VBN &quot;_" swept_VVN &quot;_" away_RL ._. 
This_DD1 is_VBZ valuable_JJ ,_, for_REX21 example_REX22 ,_, when_RRQ (_( as_CSA is_VBZ often_RR the_AT case_NN1 in_II dynamical_JJ problems_NN2 )_) A_ZZ1 is_VBZ sparse_JJ ._. 
Deflation_NN1 destroys_VVZ this_DD1 characteristic_NN1 ;_; the_AT sweeping_JJ method_NN1 under_II discussion_NN1 does_VDZ not_XX ._. 
With_IW r_ZZ1 =_FO 0_MC ,_, Equation_NN1 (_( 2.7.1.5_MC )_) becomes_VVZ &lsqb;_( formula_NN1 &rsqb;_) Hence_RR if_CS we_PPIS2 choose_VV0 co_NN1 to_TO make_VVI k1_FO zero_MC ,_, it_PPH1 will_VM contain_VVI no_AT contribution_NN1 from_II x1_FO ,_, and_CC ,_, as_CSA r_ZZ1 increases_VVZ sufficiently_RR to_II a_AT1 value_NN1 s_ZZ1 ,_, (_( 2.7.1.5_MC )_) will_VM tend_VVI to_II &lsqb;_( formula_NN1 &rsqb;_) The_AT condition_NN1 &lsqb;_( formula_NN1 &rsqb;_) means_VVZ that_CST we_PPIS2 can_VM express_VVI one_MC1 element_NN1 of_IO co_NN1 in_II31 terms_II32 of_II33 the_AT remaining_JJ n_ZZ1 1_MC1 ._. 
In_II theory_NN1 ,_, if_CS co_NN1 is_VBZ chosen_VVN in_II this_DD1 way_NN1 ,_, the_AT iterations_NN2 will_VM in_II due_JJ course_NN1 yield_NN1 (_( 2_MC )_) without_IW further_JJR attention_NN1 ._. 
In_II practice_NN1 ,_, rounding-off_JJ errors_NN2 soon_RR produces_VVZ inaccuracies_NN2 which_DDQ re-introduce_VV0 small_JJ proportions_NN2 of_IO x1_FO ,_, which_DDQ tend_VV0 to_TO grow_VVI relatively_RR rapidly_RR ._. 
It_PPH1 is_VBZ therefore_RR best_JJT to_TO apply_VVI the_AT criterion_NN1 &lsqb;_( formula_NN1 &rsqb;_) at_II each_DD1 step_NN1 ;_; i.e._REX to_TO calculate_VVI one_MC1 element_NN1 of_IO cr_NNU from_II this_DD1 criterion_NN1 when_CS the_AT others_NN2 have_VH0 been_VBN found_VVN ._. 
This_DD1 means_VVZ that_CST we_PPIS2 can_VM ignore_VVI one_MC1 row_NN1 of_IO A_ZZ1 ;_; we_PPIS2 exemplify_VV0 this_DD1 below_RL ._. 
As_CSA before_RT ,_, we_PPIS2 need_VM not_XX be_VBI completely_RR arbitrary_JJ i_ZZ1 our_APPGE choice_NN1 of_IO co_NN1 (_( though_CS it_PPH1 must_VM satisfy_VVI &lsqb;_( formula_NN1 &rsqb;_) )_) from_II (_( 1_MC1 )_) ,_, a_AT1 column_NN1 free_JJ of_IO x1_FO is_VBZ &lsqb;_( formula_NN1 &rsqb;_) with_IW co_NN1 arbitrary_JJ ._. 
But_CCB we_PPIS2 can_VV0 with_IW advantage_NN1 choose_VV0 co_NN1 to_TO be_VBI a_AT1 column_NN1 in_II the_AT previous_JJ iteration_NN1 to_II find&lsqb;formula&rsqb;1_FO etc._RA :_: this_DD1 will_VM contain_VVI mostly_RR x1_FO ,_, some_DD &lsqb;_( formula_NN1 &rsqb;_) ,_, but_CCB little_RR else_RR ;_; then_RT &lsqb;_( formula_NN1 &rsqb;_) eliminates_VVZ the_AT contribution_NN1 from_II x1_FO ._. 
Example_NN1 1_MC1 We_PPIS2 use_VV0 the_AT same_DA example_NN1 as_CSA before_RT ,_, restating_VVG for_IF convenience_NN1 what_DDQ we_PPIS2 have_VH0 already_RR found._NNU &lsqb;_( formula_NN1 &rsqb;_) Note_VV0 that_CST &lsqb;_( formula_NN1 &rsqb;_) have_VH0 been_VBN normalised_VVN to_TO give_VVI &lsqb;_( formula_NN1 &rsqb;_) ._. 
Now_RT any_DD postmultiplying_JJ column_NN1 say_VV0 a_AT1 ,_, b_ZZ1 ,_, c_ZZ1 ,_, 1_MC1 ,_, must_VM satisfy_VVI &lsqb;_( formula_NN1 &rsqb;_) In_II performing_VVG our_APPGE iterations_NN2 ,_, therefore_RR ,_, we_PPIS2 may_VM ignore_VVI completely_RR the_AT first_MD row_NN1 of_IO A_ZZ1 ;_; we_PPIS2 use_VV0 the_AT last_MD three_MC rows_NN2 to_TO find_VVI the_AT corresponding_JJ elements_NN2 in_II a_AT1 column_NN1 ,_, and_CC then_RT apply(3)_FO to_TO find_VVI the_AT top_JJ element_NN1 ._. 
For_IF our_APPGE initial_JJ column_NN1 co_NN1 ,_, we_PPIS2 again_RT choose_VV0 the_AT column_NN1 c9_FO from_II Table_NN1 2.7.1.1_MC ,_, in_II the_AT iteration_NN1 to_TO find_VVI &lsqb;_( formula_NN1 &rsqb;_) 1_MC1 etc._RA ;_; thus_RR for_IF our_APPGE present_JJ purpose_NN1 &lsqb;_( formula_NN1 &rsqb;_) Then_RT &lsqb;_( formula_NN1 &rsqb;_) and_CC &lsqb;_( formula_NN1 &rsqb;_) As_CSA it_PPH1 should_VM ,_, this_DD1 column_NN1 vanishes_VVZ when_CS premultiplied_VVN by_II &lsqb;_( formula_NN1 &rsqb;_) ._. 
We_PPIS2 divide_VV0 throughout_RL by_II the_AT last_MD element_NN1 to_TO obtain_VVI as_CSA our_APPGE new_JJ starting_NN1 column_NN1 &lsqb;_( formula_NN1 &rsqb;_) then_RT the_AT iterations_NN2 ,_, using_VVG (_( 3_MC )_) at_II each_DD1 step_NN1 ,_, begin_VV0 &lsqb;_( formula_NN1 &rsqb;_) We_PPIS2 postmultiply_RR A_ZZ1 (_( ignoring_VVG its_APPGE top_JJ row_NN1 )_) by_II the_AT column_NN1 (_( 4_MC )_) to_TO get_VVI the_AT column_NN1 of_IO three_MC numbers_NN2 on_II the_AT right_NN1 ;_; divide_VV0 by_II the_AT last_MD element_NN1 to_TO get_VVI the_AT bottom_NN1 three_MC elements_NN2 in_II the_AT next_MD column_NN1 ,_, and_CC find_VV0 the_AT top_JJ element_NN1 from_II (_( 3_MC )_) ._. 
In_II this_DD1 way_NN1 ,_, we_PPIS2 obtain_VV0 the_AT successive_JJ columns_NN2 and_CC dividing_VVG factors_NN2 in_II Table_NN1 1_MC1 ._. 
Thus_RR we_PPIS2 find_VV0 &lsqb;_( formula_NN1 &rsqb;_) ._. 
As_CSA before_RT ,_, knowing_VVG &lsqb;_( formula_NN1 &rsqb;_) 2_MC ,_, we_PPIS2 find_VV0 &lsqb;_( formula_NN1 &rsqb;_) is_VBZ proportional_JJ to_II (_( -1_MC ,_, -2,0,1_MC )_) ._. 
Normalising_NP1 :_: &lsqb;_( formula_NN1 &rsqb;_) The_AT same_DA principle_NN1 serves_VVZ for_IF the_AT third_MD root_NN1 ._. 
If_CS cr_NNU =_FO a_AT1 ,_, b_ZZ1 ,_, c_ZZ1 ,_, 1_MC1 then_JJ &lsqb;_( formula_NN1 &rsqb;_) and_CC &lsqb;_( formula_NN1 &rsqb;_) must_VM both_RR vanish_VVI ;_; we_PPIS2 find_VV0 &lsqb;_( formula_NN1 &rsqb;_) If_CS we_PPIS2 select_VV0 c3_FO from_II Table_NN1 1_MC1 as_CSA our_APPGE new_JJ co_NN1 ,_, it_PPH1 already_RR excludes_VVZ x1_FO ,_, so_CS21 that_CS22 &lsqb;_( formula_NN1 &rsqb;_) will_VM also_RR exclude_VVI &lsqb;_( formula_NN1 &rsqb;_) ._. 
This_DD1 is_VBZ &lsqb;_( formula_NN1 &rsqb;_) If_CS this_DD1 is_VBZ written_VVN 0.9333,0.0333_MC ,_, 1.0333,1_MC ,_, it_PPH1 satisfies_VVZ (_( 6_MC )_) ._. 
We_PPIS2 now_RT iterate_VV0 on_II A_ZZ1 ,_, but_CCB ignore_VV0 the_AT first_MD two_MC rows_NN2 :_: &lsqb;_( formula_NN1 &rsqb;_) We_PPIS2 leave_VV0 it_PPH1 to_II the_AT reader_NN1 to_TO show_VVI that_CST this_DD1 converges_VVZ quickly_RR to_II &lsqb;_( formula_NN1 &rsqb;_) 3_MC =_FO 5_MC ,_, x3_FO =_FO 1,0_MC ,_, -1,1_MC ._. 
In_II the_AT usual_JJ way_NN1 ,_, &lsqb;_( formula_NN1 &rsqb;_) is_VBZ found_VVN to_TO be_VBI proportional_JJ to_II 27_MC ,_, -11_MC ,_, -15.8_MC ._. 
Normalising_NP1 :_: &lsqb;_( formula_NN1 &rsqb;_) As_CSA before_RT ,_, no_AT iteration_NN1 is_VBZ needed_VVN for_IF the_AT completion_NN1 of_IO the_AT solution_NN1 ._. 
If_CS we_PPIS2 choose_VV0 the_AT last_MD column_NN1 above_RL (_( which_DDQ excludes_VVZ x1_FO and_CC &lsqb;_( formula_NN1 &rsqb;_) )_) as_II a_AT1 new_JJ co_NN1 ,_, then_RT a_AT1 column_NN1 which_DDQ also_RR excludes_VVZ x3_FO is_VBZ &lsqb;_( formula_NN1 &rsqb;_) Since_CS x1_FO ,_, &lsqb;_( formula_NN1 &rsqb;_) ,_, x3_FO are_VBR excluded_VVN ,_, this_DD1 column_NN1 is_VBZ proportional_JJ to_II x4_FO ._. 
Premultiplication_NN1 by_II a_AT1 row_NN1 of_IO A_ZZ1 gives_VVZ &lsqb;_( formula_NN1 &rsqb;_) 4_MC =_FO 2.5_MC ;_; completion_NN1 of_IO the_AT solution_NN1 is_VBZ straightforward._NNU (_( ii_MC )_) Second_MD method_NN1 Instead_II21 of_II22 sweeping_VVG our_APPGE iterated_JJ columns_NN2 at_II each_DD1 step_NN1 ,_, we_PPIS2 can_VM do_VDI it_PPH1 one_PN1 for_IF all_DB with_IW the_AT aid_NN1 of_IO a_AT1 &quot;_" sweeping_JJ matrix_NN1 &quot;_" ._. 
In_II31 view_II32 of_II33 (_( 3_MC )_) we_PPIS2 may_VM write_VVI &lsqb;_( formula_NN1 &rsqb;_) If_CS (_( 8_MC )_) is_VBZ used_VVN as_CSA postmultiplier_JJR of_IO A_ZZ1 ,_, it_PPH1 is_VBZ clear_JJ that_CST A_ZZ1 is_VBZ deflated_VVN by_II the_AT sweeping_JJ matrix_NN1 in_II (_( 8_MC )_) to_II &lsqb;_( formula_NN1 &rsqb;_) (_( note_VV0 that_CST A1_FO here_RL differs_VVZ from_II the_AT deflated_JJ form_NN1 A1_FO of_IO 2.7.2_MC )_) ._. 
If_CS A_ZZ1 is_VBZ of_IO simple_JJ form_NN1 (_( e.g._REX sparse_JJ )_) this_DD1 characteristic_NN1 may_VM be_VBI partly_RR or_CC wholly_RR destroyed_VVN in_II forming_VVG A1_FO ;_; on_II the_AT other_JJ hand_NN1 ,_, A1_FO has_VHZ a_AT1 column_NN1 of_IO zeros_NN2 ,_, and_CC thus_RR its_APPGE own_DA simplicity_NN1 ._. 
This_DD1 means_VVZ also_RR that_CST the_AT umber_NN1 of_IO the_AT head_NN1 of_IO a_AT1 column_NN1 iteration_NN1 is_VBZ always_RR multiplied_VVN by_II zero_MC ,_, an_AT1 so_RR need_VV0 not_XX be_VBI calculated_VVN till_II the_AT final_JJ step_NN1 ._. 
We_PPIS2 are_VBR thus_RR effectively_RR operating_VVG with_IW the_AT third-order_JJ submatrix_NN1 of_IO A1_FO when_RRQ its_APPGE first_MD row_NN1 and_CC column_NN1 are_VBR omitted_VVN ._. 
As_CSA before_RT ,_, we_PPIS2 may_VM find_VVI a_AT1 suitable_JJ initial_JJ column_NN1 such_II21 as_II22 (_( 4_MC )_) ._. 
We_PPIS2 then_RT perform_VV0 the_AT iteration_NN1 ,_, omitting_VVG the_AT first_MD element_NN1 of_IO (_( 4_MC )_) :_: &lsqb;_( formula_NN1 &rsqb;_) The_AT numbers_NN2 produced_VVN in_II this_DD1 way_NN1 reproduced_VVD exactly_RR those_DD2 of_IO the_AT last_MD three_MC elements_NN2 in_II the_AT columns_NN2 of_IO Table_NN1 1_MC1 ,_, as_II31 well_II32 as_II33 the_AT dividing_JJ factors_NN2 ;_; they_PPHS2 are_VBR of_RR21 course_RR22 exactly_RR the_AT same_DA calculations_NN2 ,_, but_CCB with_IW the_AT matrices_NN2 associated_VVN differently_RR ._. 
At_II the_AT conclusion_NN1 ,_, the_AT top_JJ element_NN1 of_IO the_AT final_JJ column_NN1 is_VBZ found_VVN by_II use_NN1 of_IO the_AT first_MD row_NN1 of_IO A1_FO ._. 
We_PPIS2 must_VM emphasise_VVI ,_, however_RR ,_, that_CST the_AT row_NN1 vectors_NN2 A1_FO are_VBR not_XX &lsqb;_( formula_NN1 &rsqb;_) ,_, &lsqb;_( formula_NN1 &rsqb;_) ._. 
Hence_RR ,_, when_RRQ &lsqb;_( formula_NN1 &rsqb;_) 2_MC has_VHZ been_VBN found_VVN ,_, we_PPIS2 revert_VV0 to_II A_ZZ1 itself_PPX1 to_TO find_VVI &lsqb;_( formula_NN1 &rsqb;_) ._. 
For_IF the_AT next_MD root_NN1 ,_, we_PPIS2 use_VV0 another_DD1 sweeping_JJ matrix_NN1 based_VVN on_II the_AT second_MD part_NN1 of_IO (_( 5_MC )_) :_: &lsqb;_( formula_NN1 &rsqb;_) If_CS this_DD1 square_JJ matrix_NN1 is_VBZ used_VVN to_II postmultiply_RR A1_FO we_PPIS2 find_VV0 &lsqb;_( formula_NN1 &rsqb;_) We_PPIS2 now_RT iterate_VV0 on_II the_AT submatrix_NN1 &lsqb;_( formula_NN1 &rsqb;_) to_TO determine_VVI the_AT last_MD two_MC elements_NN2 of_IO x3_FO ;_; we_PPIS2 leave_VV0 the_AT completion_NN1 of_IO the_AT solution_NN1 to_II the_AT reader._NNU ,_, 2.7.4_MC Shifting_JJ and_CC inverse_JJ iteration_NN1 Theorem_NN1 IV_MC of_IO 1.22_MC noted_VVD that_CST ,_, in_II31 view_II32 of_II33 the_AT identity_NN1 &lsqb;_( formula_NN1 &rsqb;_) addition_NN1 of_IO a_AT1 quantity_NN1 ge_NN1 to_II each_DD1 of_IO the_AT diagonal_JJ elements_NN2 of_IO A_ZZ1 implies_VVZ the_AT addition_NN1 of_IO ge_NN1 to_II each_DD1 of_IO the_AT eigenvalues_NN2 of_IO A._NP1 We_PPIS2 note_VV0 here_RL ,_, additionally_RR ,_, that_CST the_AT constituent_NN1 matrices_NN2 zii_NN2 of_IO A_ZZ1 are_VBR unaltered_JJ by_II such_DA a_AT1 change_NN1 ._. 
Sylvester_NP1 's_GE expansion_NN1 gives_VVZ (_( see_VV0 1.18_MC )_) &lsqb;_( formula_NN1 &rsqb;_) whence_RRQ &lsqb;_( formula_NN1 &rsqb;_) The_AT device_NN1 of_IO adding_VVG ge_NN1 in_II this_DD1 way_NN1 is_VBZ known_VVN as_II &quot;_" shifting_JJ &quot;_" ,_, and_CC has_VHZ a_AT1 variety_NN1 of_IO uses_NN2 ._. 
For_REX21 example_REX22 ,_, with_IW hindsight_NN1 ,_, we_PPIS2 know_VV0 that_CST the_AT eigenvalues_NN2 of_IO A_ZZ1 in_II (_( 2.6.1.1_MC )_) are_VBR 25_MC ,_, 12.5_MC ,_, 5_MC and_CC 2.5._MC the_AT rapidity_NN1 of_IO convergence_NN1 of_IO the_AT initial_JJ power_NN1 solution_NN1 depends_VVZ principally_RR on_II the_AT relative_JJ separation_NN1 of_IO the_AT eigenvalues_NN2 25_MC and_CC 12.5_MC ,_, i.e._REX 2:1_MC ._. 
If_CS we_PPIS2 reduce_VV0 all_DB eigenvalues_NN2 by_II 7.5_MC ,_, they_PPHS2 become_VV0 17.5_MC ,_, 5_MC ,_, 2.5_MC ,_, 5_MC and_CC the_AT relative_JJ separation_NN1 is_VBZ 31/2:1_MC which_DDQ yields_VVZ much_DA1 quicker_JJR convergence_NN1 ._. 
Note_VV0 that_CST this_DD1 is_VBZ the_AT optimum_JJ :_: the_AT second_MD and_CC fourth_MD eigenvalues_NN2 now_RT have_VH0 equal_JJ moduli_NN2 ._. 
To_TO select_VVI the_AT optimum_JJ requires_VVZ accurate_JJ knowledge_NN1 of_IO the_AT eigenvalues_NN2 ,_, which_DDQ at_II the_AT outset_NN1 of_IO a_AT1 calculation_NN1 is_VBZ of_RR21 course_RR22 not_XX available_JJ ;_; but_CCB if_CS one_PN1 has_VHZ a_AT1 general_JJ idea_NN1 of_IO the_AT disposition_NN1 of_IO the_AT eigenvalues_NN2 ,_, shifting_NN1 can_VM save_VVI much_DA1 time_NNT1 ._. 
The_AT principle_NN1 use_NN1 of_IO shifting_NN1 ,_, however_RR ,_, lies_VVZ in_II inverse_JJ iteration_NN1 ._. 
Suppose_VV0 ,_, for_REX21 example_REX22 ,_, that_CST the_AT matrix_NN1 A_ZZ1 is_VBZ the_AT system_NN1 matrix_NN1 of_IO a_AT1 vibrating_JJ mechanical_JJ system_NN1 ,_, and_CC suppose_VV0 further_RRR that_CST this_DD1 system_NN1 is_VBZ excited_VVN by_II a_AT1 forcing_NN1 vibration_NN1 of_IO frequency_NN1 corresponding_VVG to_II &lsqb;_( formula_NN1 &rsqb;_) =_FO &lsqb;_( formula_NN1 &rsqb;_) ._. 
It_PPH1 is_VBZ desired_VVN to_TO determine_VVI the_AT eigenvalue_NN1 (_( and_CC its_APPGE vectors_NN2 )_) nearest_II21 to_II22 &lsqb;_( formula_NN1 &rsqb;_) in_BCL21 order_BCL22 to_TO study_VVI the_AT response_NN1 ._. 
To_TO approach_VVI this_DD1 problem_NN1 ,_, we_PPIS2 first_MD note_NN1 that_CST Sylvester_NP1 's_GE expansion_NN1 for_IF negative_JJ powers_NN2 of_IO a_AT1 matrix_NN1 B_ZZ1 gives_VVZ &lsqb;_( formula_NN1 &rsqb;_) It_PPH1 follows_VVZ that_CST the_AT eigenvalue_NN1 of_IO B_ZZ1 with_IW the_AT smallest_JJT modulus_NN1 now_RT becomes_VVZ the_AT eigenvalue_NN1 of_IO B-1_FO with_IW largest_JJT modulus_NN1 ._. 
Hence_RR ,_, to_TO solve_VVI our_APPGE problem_NN1 ,_, we_PPIS2 first_MD evaluate_VV0 B_ZZ1 =_FO A_ZZ1 -&lsqb;formula&rsqb;_JJ the_AT eigenvalue_NN1 of_IO A_ZZ1 nearest_II21 to_II22 &lsqb;_( formula_NN1 &rsqb;_) now_RT becomes_VVZ the_AT eigenvalue_NN1 of_IO B_ZZ1 nearest_II zero_MC ,_, i.e._REX of_IO smallest_JJT modulus_NN1 ._. 
(_( Note_VV0 that_CST A_ZZ1 ,_, B_ZZ1 and_CC B-1_FO all_RR have_VH0 the_AT same_DA constituent_NN1 matrices_NN2 zii_NN2 ._. )_) 
We_PPIS2 therefore_RR invert_VV0 B_ZZ1 and_CC iterate_VV0 on_II B-1_FO as_RR21 usual_RR22 to_TO obtain_VVI our_APPGE solution_NN1 ._. 
Example_NN1 1_MC1 Once_RR21 more_RR22 we_PPIS2 choose_VV0 the_AT matrix_NN1 A_ZZ1 defined_VVN in_II (_( 2.6.1.1_MC )_) and_CC used_VVN in_II previous_JJ examples_NN2 ._. 
We_PPIS2 also_RR let_VV0 &lsqb;_( formula_NN1 &rsqb;_) =_FO 5.5_MC ._. 
Then_RT &lsqb;_( formula_NN1 &rsqb;_) If_CS we_PPIS2 invert_VV0 this_DD1 ,_, e.g._REX by_II the_AT methods_NN2 of_IO 2.2_MC ,_, we_PPIS2 obtain_VV0 &lsqb;_( formula_NN1 &rsqb;_) We_PPIS2 postmultiply_RR this_DD1 by_II an_AT1 arbitrary_JJ column_NN1 co_NN1 :_: here_RL we_PPIS2 choose_VV0 co_NN1 =_FO e1_FO and_CC iterate_VV0 as_RR21 usual_RR22 ._. 
The_AT successive_JJ columns_NN2 are_VBR given_VVN in_II Table_NN1 1_MC1 ._. 
The_AT iterations_NN2 converge_VV0 rapidly_RR to_II x_ZZ1 =_FO 1,0_MC ,_, 1,1_MC with_IW the_AT eigenvalue_NN1 of_IO B-1_FO equal_JJ to_II -2_MC ._. 
Hence_RR the_AT eigenvalue_NN1 of_IO B_ZZ1 is_VBZ 0.5_MC an_AT1 of_IO A_ZZ1 is_VBZ 5.5_MC 0.5_MC =_FO 5_MC ,_, in_II31 accordance_II32 with_II33 results_NN2 obtained_VVN earlier_RRR for_IF &lsqb;_( formula_NN1 &rsqb;_) 3_MC and_CC x3_FO ._. 
Note_VV0 that_CST ,_, since_CS the_AT eigenvalues_NN2 of_IO A_ZZ1 are_VBR in_II fact_NN1 25_MC ,_, 12.5_MC ,_, 5_MC ,_, 2.5_MC ,_, those_DD2 of_IO B_ZZ1 are_VBR 19.5_MC ,_, 7_MC ,_, 0.5_MC ,_, 8_MC ,_, and_CC of_IO B-1_FO therefore_RR 0.0513_MC ,_, 0.1429_MC ,_, 2_MC ,_, 0.124_MC ._. 
The_AT relative_JJ separation_NN1 of_IO the_AT two_MC roots_NN2 of_IO largest_JJT moduli_NN2 is_VBZ here_RL 14:1_MC ._. 
Clearly_RR ,_, the_AT method_NN1 is_VBZ not_XX restricted_VVN to_II mechanical_JJ systems_NN2 :_: by_II its_APPGE means_NN we_PPIS2 can_VM study_VVI the_AT eigenvalues_NN2 and_CC vectors_NN2 of_IO any_DD matrix_NN1 A_ZZ1 which_DDQ lie_VV0 near_II any_DD given_JJ value_NN1 &lsqb;_( formula_NN1 &rsqb;_) o_ZZ1 of_IO &lsqb;_( formula_NN1 &rsqb;_) ._. 
&lsqb;_( formula_NN1 &rsqb;_) 2.7.5_MC Confluent_JJ eigenvalues_NN2 So_RG far_RR ,_, we_PPIS2 have_VH0 restricted_VVN our_APPGE attention_NN1 to_II unrepeated_JJ real_JJ roots_NN2 of_IO the_AT characteristic_JJ equation_NN1 belonging_VVG to_II a_AT1 matrix_NN1 A_ZZ1 ;_; we_PPIS2 now_RT consider_VV0 the_AT case_NN1 of_IO two_MC equal_JJ real_JJ roots_NN2 ._. 
Multiple_JJ roots_NN2 are_VBR in_II practice_NN1 very_JJ rate_NN1 (_( except_CS perhaps_RR for_IF mechanical_JJ systems_NN2 moving_VVG freely_RR in_II space_NN1 ,_, when_CS multiple_JJ zero_NN1 roots_NN2 can_VM occur_VVI :_: but_CCB these_DD2 have_VH0 a_AT1 special_JJ simplicity_NN1 )_) ._. 
The_AT treatment_NN1 for_IF two_MC roots_NN2 is_VBZ readily_RR extended_VVN to_II the_AT multiple_JJ case_NN1 if_CS required_VVN :_: even_RR for_IF two_MC roots_NN2 ,_, however_RR ,_, we_PPIS2 require_VV0 to_TO examine_VVI two_MC cases_NN2 (_( see_VV0 1.21_MC )_) ._. 
We_PPIS2 assume_VV0 the_AT roots_NN2 to_TO have_VHI dominant_JJ moduli_NN2 ,_, any_DD roots_NN2 of_IO greater_JJR modulus_NN1 having_VHG been_VBN removed_VVN by_II deflation_NN1 etc._RA (_( i_ZZ1 )_) Characteristic_JJ matrix_NN1 doubly_RR degenerate_JJ When_CS a_AT1 repeated_JJ eigenvalue_NN1 &lsqb;_( formula_NN1 &rsqb;_) 1_MC1 makes_VVZ the_AT characteristic_JJ matrix_NN1 doubly_RR degenerate_JJ ,_, we_PPIS2 know_VV0 (_( see_VV0 (_( 1.21.14_MC )_) )_) that_CST the_AT Sylvester_NP1 expansion_NN1 of_IO the_AT system_NN1 matrix_NN1 A_ZZ1 is_VBZ &lsqb;_( formula_NN1 &rsqb;_) It_PPH1 follows_VVZ that_CST we_PPIS2 can_VM use_VVI the_AT power_NN1 method_NN1 to_TO find_VVI &lsqb;_( formula_NN1 &rsqb;_) 1_MC1 ;_; postmultiplication_NN1 by_II co_NN1 gives_VVZ in_II the_AT limit_NN1 ,_, when_CS r_ZZ1 is_VBZ sufficiently_RR large_JJ ,_, say_VV0 s_ZZ1 ,_, &lsqb;_( formula_NN1 &rsqb;_) We_PPIS2 thus_RR find_VV0 &lsqb;_( formula_NN1 &rsqb;_) 1_MC1 without_IW difficulty_NN1 ;_; but_CCB the_AT determination_NN1 of_IO z11_FO +_FO z22_FO requires_VVZ further_JJR consideration_NN1 ._. 
Now_RT &lsqb;_( formula_NN1 &rsqb;_) and_CC &lsqb;_( formula_NN1 &rsqb;_) is_VBZ thus_RR ,_, in_RR21 general_RR22 a_AT1 linear_JJ combination_NN1 of_IO 1_MC1 ,_, &lsqb;_( formula_NN1 &rsqb;_) ._. 
At_II this_DD1 point_NN1 ,_, the_AT iteration_NN1 has_VHZ given_VVN no_AT indication_NN1 that_CST a_AT1 double_JJ root_NN1 is_VBZ involved_VVN ;_; however_RR ,_, this_DD1 becomes_VVZ apparent_JJ when_CS we_PPIS2 seek_VV0 the_AT row_NN1 vector_NN1 corresponding_VVG to_II cs_FO :_: for_IF all_DB first_MD minors_NN2 of_IO &lsqb;_( formula_NN1 &rsqb;_) 1I_FO A_ZZ1 vanish_VV0 and_CC we_PPIS2 can_VM not_XX find_VVI a_AT1 unique_JJ row_NN1 vector_NN1 ._. 
To_TO proceed_VVI ,_, it_PPH1 is_VBZ simplest_JJT to_TO generalise_VVI (_( 2.6.4_MC )_) by_II partitioning_NN1 off_II the_AT last_MD two_MC rows_NN2 and_CC columns_NN2 of_IO &lsqb;_( formula_NN1 &rsqb;_) 1I_FO A_ZZ1 ,_, thus_RR :_: &lsqb;_( formula_NN1 &rsqb;_) Here_RL &lsqb;_( formula_NN1 &rsqb;_) &lsqb;_( formula_NN1 &rsqb;_) is_VBZ a_AT1 square_JJ submatrix_NN1 of_IO order_NN1 &lsqb;_( formula_NN1 &rsqb;_) of_IO order_NN1 2_MC ;_; &lsqb;_( formula_NN1 &rsqb;_) has_VHZ two_MC columns_NN2 and_CC &lsqb;_( formula_NN1 &rsqb;_) has_VHZ two_MC rows_NN2 ._. 
Then_RT &lsqb;_( formula_NN1 &rsqb;_) These_DD2 algebraic_JJ equations_NN2 are_VBR solved_VVN for_IF &lsqb;_( formula_NN1 &rsqb;_) ;_; we_PPIS2 now_RT have_VH0 two_MC independent_JJ columns_NN2 &lsqb;_( formula_NN1 &rsqb;_) which_DDQ annihilate_VV0 &lsqb;_( formula_NN1 &rsqb;_) 1I_FO A_ZZ1 when_CS used_VVN as_CSA postmultipliers_NN2 ;_; and_CC correspondingly_RR for_IF the_AT rows_NN2 &lsqb;_( formula_NN1 &rsqb;_) ._. 
We_PPIS2 now_RT normalise_VV0 them_PPHO2 ;_; we_PPIS2 evaluate_VV0 &lsqb;_( formula_NN1 &rsqb;_) Then_RT m_ZZ1 is_VBZ a_AT1 square_JJ matrix_NN1 of_IO order_NN1 2_MC and_CC ,_, for_REX21 example_REX22 ,_, &lsqb;_( formula_NN1 &rsqb;_) are_VBR normalised_JJ rows_NN2 and_CC columns_NN2 ._. 
Finally_RR &lsqb;_( formula_NN1 &rsqb;_) Note_VV0 that_CST only_RR the_AT sum_NN1 &lsqb;_( formula_NN1 &rsqb;_) is_VBZ unique_JJ ._. 
If_CS they_PPHS2 are_VBR individually_RR expressed_VVN as_CSA &lsqb;_( formula_NN1 &rsqb;_) and_CC &lsqb;_( formula_NN1 &rsqb;_) then_RT their_APPGE sum_NN1 can_VM be_VBI written_VVN &lsqb;_( formula_NN1 &rsqb;_) where_RRQ M_ZZ1 is_VBZ an_AT1 arbitrary_JJ non-singular_JJ matrix_NN1 of_IO order_NN1 2_MC ._. 
This_DD1 replaces_VVZ x1_FO ,_, &lsqb;_( formula_NN1 &rsqb;_) with_IW two_MC different_JJ linear_JJ combinations_NN2 (_( x1_FO ,_, &lsqb;_( formula_NN1 &rsqb;_) )_) M_NN1 of_IO them_PPHO2 ,_, with_IW corresponding_JJ changes_NN2 for_IF &lsqb;_( formula_NN1 &rsqb;_) ._. 
It_PPH1 will_VM be_VBI observed_VVN that_CST in_II this_DD1 approach_NN1 we_PPIS2 have_VH0 not_XX used_VVN cs_FO (_( which_DDQ we_PPIS2 have_VH0 already_RR found_VVN )_) ;_; its_APPGE introduction_NN1 would_VM spoil_VVI the_AT simplicity_NN1 of_IO (_( 2_MC )_) ._. 
When_CS x1_FO ,_, &lsqb;_( formula_NN1 &rsqb;_) have_VH0 been_VBN found_VVN ,_, it_PPH1 is_VBZ possible_JJ to_TO check_VVI that_CST cs_FO is_VBZ a_AT1 linear_JJ combination_NN1 of_IO them._NNU ,_, Example_NN1 1_MC1 Consider_VV0 the_AT matrix_NN1 &lsqb;_( formula_NN1 &rsqb;_) It_PPH1 is_VBZ assumed_VVN that_CST at_II this_DD1 stage_NN1 we_PPIS2 have_VH0 no_AT knowledge_NN1 of_IO its_APPGE eigenvalues_NN2 ._. 
If_CS we_PPIS2 choose_VV0 &lsqb;_( formula_NN1 &rsqb;_) and_CC iterate_VV0 in_II the_AT usual_JJ way_NN1 ,_, we_PPIS2 find_VV0 quite_RG quickly_RR that_DD1 &lsqb;_( formula_NN1 &rsqb;_) Normally_RR ,_, we_PPIS2 would_VM choose_VVI &lsqb;_( formula_NN1 &rsqb;_) However_RR ,_, if_CS we_PPIS2 attempt_VV0 to_TO use_VVI (_( 2.6.4_MC )_) to_TO find_VVI &lsqb;_( formula_NN1 &rsqb;_) ,_, it_PPH1 at_RR21 once_RR22 emerges_VVZ that_CST all_DB first_MD minors_NN2 of_IO &lsqb;_( formula_NN1 &rsqb;_) 1I-_NN1 A_ZZ1 vanish_VV0 ,_, indicating_VVG that_CST &lsqb;_( formula_NN1 &rsqb;_) 1_MC1 =_FO 12.5_MC is_VBZ a_AT1 repeated_JJ root_NN1 ._. 
We_PPIS2 therefore_RR apply_VV0 (_( 2_MC )_) with_IW &lsqb;_( formula_NN1 &rsqb;_) This_DD1 gives_VVZ for_IF the_AT solution_NN1 of_IO (_( 2_MC )_) &lsqb;_( formula_NN1 &rsqb;_) Since_CS the_AT columns_NN2 and_CC rows_NN2 are_VBR arbitrary_JJ to_II a_AT1 scalar_JJ multiplier_NN1 ,_, we_PPIS2 may_VM now_RT write_VVI &lsqb;_( formula_NN1 &rsqb;_) (_( for_IF simplicity_NN1 of_IO exposition_NN1 we_PPIS2 have_VH0 multiplied_VVN the_AT first_MD column_NN1 on_II the_AT right_NN1 by_II 3_MC )_) ._. 
The_AT product_NN1 of_IO the_AT numerical_JJ matrices_NN2 in_II (_( 8_MC )_) ,_, in_II that_DD1 order_NN1 ,_, is_VBZ (_( see_VV0 (_( 3_MC )_) )_) &lsqb;_( formula_NN1 &rsqb;_) If_CS we_PPIS2 premultiply_RR &lsqb;_( formula_NN1 &rsqb;_) we_PPIS2 ca_VM finally_RR write_VVI &lsqb;_( formula_NN1 &rsqb;_) These_DD2 enable_VV0 us_PPIO2 to_TO choose_VVI ,_, if_CS they_PPHS2 are_VBR required_VVN ,_, &lsqb;_( formula_NN1 &rsqb;_) but_CCB ,_, following_RA (_( 5_MC )_) ,_, we_PPIS2 could_VM equally_RR well_RR use_VVI any_DD two_MC different_JJ linear_JJ combinations_NN2 o_ZZ1 x1_FO ,_, &lsqb;_( formula_NN1 &rsqb;_) ,_, represented_VVN by_II (_( x1_FO ,_, &lsqb;_( formula_NN1 &rsqb;_) )_) M_ZZ1 provided_CS these_DD2 are_VBR associated_VVN with_IW &lsqb;_( formula_NN1 &rsqb;_) ._. ,_, 
/div4._FU (_( ii_MC )_) Characteristic_JJ matrix_NN1 simply_RR degenerate_JJ This_DD1 case_NN1 is_VBZ rather_RG more_RGR difficult_JJ :_: the_AT difficulty_NN1 derives_VVZ from_II the_AT appearance_NN1 of_IO a_AT1 unit_NN1 in_II the_AT superdiagonal_JJ of_IO the_AT spectral_JJ matrix_NN1 of_IO the_AT given_JJ matrix_NN1 A._NNU For_IF simplicity_NN1 of_IO exposition_NN1 we_PPIS2 assume_VV0 here_RL that_CST A_ZZ1 is_VBZ of_IO order_NN1 4_MC :_: extension_NN1 to_II larger_JJR matrices_NN2 is_VBZ obvious_JJ ._. 
We_PPIS2 know_VV0 (_( see_VV0 1.21_MC )_) that_CST &lsqb;_( formula_NN1 &rsqb;_) ,_, &lsqb;_( formula_NN1 &rsqb;_) so_CS21 that_CS22 &lsqb;_( formula_NN1 &rsqb;_) on_II expansion_NN1 of_IO the_AT triple_JJ product_NN1 ._. 
We_PPIS2 know_VV0 further_RRR that_DD1 ,_, in_II31 view_II32 of_II33 the_AT properties_NN2 of_IO the_AT constituent_NN1 matrices_NN2 zij_VV0 ,_, &lsqb;_( formula_NN1 &rsqb;_) When_RRQ &lsqb;_( formula_NN1 &rsqb;_) is_VBZ absent_JJ ,_, this_DD1 reverts_VVZ to_II the_AT simpler_JJR case_NN1 dealt_VVN with_IW under_II (_( i_ZZ1 )_) ._. 
Our_APPGE task_NN1 now_RT is_VBZ ,_, given_VVN A_ZZ1 ,_, to_TO resolve_VVI it_PPH1 numerically_RR into_II one_MC1 of_IO the_AT forms_NN2 (_( 12_MC )_) ,_, i.e._REX to_TO find_VVI &lsqb;_( formula_NN1 &rsqb;_) 1_MC1 etc_RA ._. 
Sine_FW ,_, a_JJ21 priori_JJ22 ,_, we_PPIS2 do_VD0 not_XX know_VVI the_AT nature_NN1 of_IO A_ZZ1 ,_, we_PPIS2 should_VM approach_VVI this_DD1 task_NN1 as_RR21 usual_RR22 by_II iteration_NN1 with_IW A_ZZ1 on_II an_AT1 arbitrary_JJ column_NN1 co_NN1 ._. 
If_CS we_PPIS2 write_VV0 &lsqb;_( formula_NN1 &rsqb;_) then_RT postmultiplication_NN1 of_IO (_( 13_MC )_) by_II co_NN1 yields_NN2 &lsqb;_( formula_NN1 &rsqb;_) When_RRQ the_AT iterations_NN2 have_VH0 proceeded_VVN so_RG far_RR ,_, say_VV0 when_RRQ r_ZZ1 =_FO s_ZZ1 ,_, that_CST contributions_NN2 from_II &lsqb;_( formula_NN1 &rsqb;_) 3_MC ,_, &lsqb;_( formula_NN1 &rsqb;_) 4_MC are_VBR insignificant_JJ and_CC may_VM be_VBI neglected_VVN ,_, then_RT &lsqb;_( formula_NN1 &rsqb;_) Previously_RR ,_, with_IW &lsqb;_( formula_NN1 &rsqb;_) absent_JJ ,_, at_II this_DD1 stage_NN1 &lsqb;_( formula_NN1 &rsqb;_) equalled_VVD &lsqb;_( formula_NN1 &rsqb;_) ._. 
Now_RT ,_, owing_II21 to_II22 the_AT last_MD term_NN1 ,_, this_DD1 is_VBZ no_RR21 longer_RR22 true_JJ and_CC successive_JJ columns_NN2 tend_VV0 to_TO show_VVI a_AT1 near_JJ arithmetical_JJ progression_NN1 in_II their_APPGE elements_NN2 ._. 
It_PPH1 is_VBZ true_JJ that_CST ,_, when_CS s_ZZ1 is_VBZ very_RG large_JJ &lsqb;_( formula_NN1 &rsqb;_) the_AT columns_NN2 tend_VV0 ultimately_RR to_II x1_FO (_( last_MD term_NN1 in_II cs_FO dominant_JJ )_) ,_, but_CCB the_AT number_NN1 of_IO iterations_NN2 required_VVN is_VBZ very_RG large_JJ ._. 
However_RR ,_, in_II31 place_II32 of_II33 the_AT simple_JJ &lsqb;_( formula_NN1 &rsqb;_) we_PPIS2 can_VM drive_VVI from_II (_( 15_MC )_) the_AT relation_NN1 &lsqb;_( formula_NN1 &rsqb;_) In_II practice_NN1 ,_, therefore_RR ,_, if_CS convergence_NN1 seems_VVZ very_RG slow_JJ ,_, we_PPIS2 evaluate_VV0 three_MC successive_JJ columns_NN2 fully_RR ,_, omitting_VVG the_AT reduction_NN1 of_IO a_AT1 homologous_JJ element_NN1 to_II unity_NN1 ,_, and_CC solve_VV0 the_AT quadratic_JJ &lsqb;_( formula_NN1 &rsqb;_) for_IF at_RR21 least_RR22 two_MC homologous_JJ elements_NN2 ._. 
If_CS they_PPHS2 have_VH0 a_AT1 common_JJ root_NN1 ,_, we_PPIS2 test_VV0 the_AT remaining_JJ quadratics_NN2 to_TO see_VVI if_CSW they_PPHS2 are_VBR satisfied_VVN by_II this_DD1 common_JJ root_NN1 ._. 
If_CS they_PPHS2 re_II ,_, then_RT the_AT root_NN1 is_VBZ &lsqb;_( formula_NN1 &rsqb;_) 1_MC1 and_CC the_AT iterations_NN2 have_VH0 proceeded_VVN far_RR enough_RR (_( r_ZZ1 =_FO s_ZZ1 )_) for_IF only_JJ terms_NN2 in_II &lsqb;_( formula_NN1 &rsqb;_) 1_MC1 to_TO remain_VVI ._. 
Knowing_VVG &lsqb;_( formula_NN1 &rsqb;_) 1_MC1 we_PPIS2 use_VV0 (_( 15_MC )_) to_TO evaluate_VVI &lsqb;_( formula_NN1 &rsqb;_) giving_VVG us_PPIO2 a_AT1 column_NN1 proportional_JJ to_II x1_FO ._. 
Knowing_VVG &lsqb;_( formula_NN1 &rsqb;_) 1_MC1 ,_, we_PPIS2 now_RT form_VV0 the_AT matrix_NN1 &lsqb;_( formula_NN1 &rsqb;_) 1I_FO A_ZZ1 ;_; if_CS we_PPIS2 put_VV0 r_ZZ1 =_FO 0,1_MC in_II (_( 13_MC )_) ,_, multiply_VV0 the_AT first_MD by_II &lsqb;_( formula_NN1 &rsqb;_) 1_MC1 and_CC then_RT subtract_VV0 the_AT second_NNT1 :_: &lsqb;_( formula_NN1 &rsqb;_) Premultiplication_NN1 of_IO this_DD1 by_II &lsqb;_( formula_NN1 &rsqb;_) ,_, or_CC postmultiplication_NN1 by_II x1_FO ,_, annihilates_VVZ it_PPH1 ;_; none_PN of_IO the_AT other_JJ vectors_NN2 does_VDZ so_RR :_: the_AT matrix_NN1 is_VBZ rendered_VVN simply_RR degenerate_JJ by_II the_AT presence_NN1 of_IO z12_FO ._. 
We_PPIS2 may_VM ,_, however_RR ,_, use_VVI (_( 2.6.4_MC )_) to_TO find_VVI &lsqb;_( formula_NN1 &rsqb;_) ,_, we_PPIS2 then_RT known_VVN &lsqb;_( formula_NN1 &rsqb;_) 1_MC1 and_CC &lsqb;_( formula_NN1 &rsqb;_) subject_II21 to_II22 scalar_JJ multipliers_NN2 ._. 
To_TO find_VVI &lsqb;_( formula_NN1 &rsqb;_) we_PPIS2 revert_VV0 to_II (_( 11_MC )_) ,_, which_DDQ yield_VV0 ,_, inter_RR21 alia_RR22 ,_, &lsqb;_( formula_NN1 &rsqb;_) or_CC &lsqb;_( formula_NN1 &rsqb;_) Since_CS &lsqb;_( formula_NN1 &rsqb;_) 1I_FO A_ZZ1 is_VBZ simply_RR degenerate_JJ ,_, the_AT last_MD equations_NN2 provide_VV0 ,_, in_RR21 general_RR22 ,_, n_ZZ1 1_MC1 independent_JJ linear_JJ equations_NN2 for_IF &lsqb;_( formula_NN1 &rsqb;_) (_( and_CC &lsqb;_( formula_NN1 &rsqb;_) )_) ._. 
However_RR ,_, one_MC1 element_NN1 in_II each_DD1 of_IO these_DD2 vectors_NN2 may_VM be_VBI assigned_VVN arbitrarily_RR ._. 
For_IF ,_, since_CS (_( &lsqb;_( formula_NN1 &rsqb;_) 1I_FO A_ZZ1 )_) x1_FO vanishes_VVZ ,_, the_AT equation_NN1 for_IF &lsqb;_( formula_NN1 &rsqb;_) is_VBZ also_RR satisfied_VVN by_II &lsqb;_( formula_NN1 &rsqb;_) ,_, where_CS q_ZZ1 is_VBZ arbitrary_JJ ;_; so_RR ,_, for_REX21 example_REX22 ,_, if_CS we_PPIS2 choose_VV0 a_AT1 non-zero_JJ element_NN1 in_II &lsqb;_( formula_NN1 &rsqb;_) ,_, the_AT homologous_JJ element_NN1 in_II &lsqb;_( formula_NN1 &rsqb;_) may_VM be_VBI made_VVN zero_NN1 by_II appropriate_JJ choice_NN1 of_IO 1_MC1 ._. 
Similar_JJ considerations_NN2 apply_VV0 for_IF &lsqb;_( formula_NN1 &rsqb;_) ._. 
We_PPIS2 are_VBR thus_RR able_JK to_TO solve_VVI (_( 19_MC )_) for_IF &lsqb;_( formula_NN1 &rsqb;_) and_CC &lsqb;_( formula_NN1 &rsqb;_) ._. 
Having_VHG done_VDN this_DD1 ,_, we_PPIS2 now_RT require_VV0 to_TO normalise_VVI :_: we_PPIS2 evaluate_VV0 &lsqb;_( formula_NN1 &rsqb;_) and_CC we_PPIS2 may_VM then_RT choose_VVI our_APPGE vectors_NN2 as_CSA ,_, for_REX21 example_REX22 ,_, &lsqb;_( formula_NN1 &rsqb;_) ,_, &lsqb;_( formula_NN1 &rsqb;_) and_CC the_AT pair_NN &lsqb;_( formula_NN1 &rsqb;_) which_DDQ we_PPIS2 now_RT call_VV0 &lsqb;_( formula_NN1 &rsqb;_) ._. 
We_PPIS2 have_VH0 now_RT obtained_VVN &lsqb;_( formula_NN1 &rsqb;_) 1_MC1 the_AT vectors_NN2 &lsqb;_( formula_NN1 &rsqb;_) and_CC &lsqb;_( formula_NN1 &rsqb;_) ,_, and_CC the_AT auxiliary_JJ vectors_NN2 &lsqb;_( formula_NN1 &rsqb;_) and_CC &lsqb;_( formula_NN1 &rsqb;_) ;_; we_PPIS2 are_VBR therefore_RR in_II a_AT1 position_NN1 to_TO deflate_VVI A_ZZ1 (_( see_VV0 (_( 12_MC )_) )_) by_II removing_VVG all_DB contributions_NN2 from_II these_DD2 quantities_NN2 ._. 
The_AT solution_NN1 is_VBZ then_RT completed_VVN in_II the_AT usual_JJ way_NN1 ._. 
Example_NN1 2_MC We_PPIS2 require_VV0 to_TO find_VVI the_AT modal_NN1 and_CC spectral_JJ matrices_NN2 of_IO &lsqb;_( formula_NN1 &rsqb;_) We_PPIS2 begin_VV0 by_II iterating_VVG on_II co_NN1 ._. 
If_CS we_PPIS2 choose_VV0 co_NN1 =_FO e4_FO ,_, then_RT after_CS about_RG 18_MC steps_NN2 ,_, four_MC successive_JJ columns_NN2 are_VBR &lsqb;_( formula_NN1 &rsqb;_) These_DD2 show_VV0 very_RG slow_JJ convergence_NN1 ;_; indeed_RR ,_, the_AT elements_NN2 show_VV0 a_AT1 near-arithmetical_JJ progression_NN1 ._. 
We_PPIS2 therefore_RR choose_VV0 ,_, say_VV0 ,_, the_AT first_MD of_IO these_DD2 columns_NN2 and_CC without_IW reducing_VVG the_AT last_MD element_NN1 to_II unity_NN1 ,_, calculate_VV0 the_AT next_MD two_MC columns_NN2 :_: &lsqb;_( formula_NN1 &rsqb;_) We_PPIS2 choose_VV0 ,_, say_VV0 ,_, the_AT first_MD and_CC last_MD elements_NN2 of_IO these_DD2 and_CC solve_VV0 &lsqb;_( formula_NN1 &rsqb;_) The_AT common_JJ root_NN1 12.5_MC also_RR satisfies_VVZ the_AT middle_NN1 two_MC quadratics_NN2 ;_; hence_RR we_PPIS2 find_VV0 &lsqb;_( formula_NN1 &rsqb;_) 1_MC1 =_FO 12.5_MC ._. 
We_PPIS2 now_RT use_VV0 (_( 17_MC )_) and_CC find_VV0 2.2875_MC ,_, 4.6875_MC ,_, 7.5_MC ,_, 13.125_MC 12.5_MC 0.165_MC ,_, 0.355_MC ,_, 0.57,1_MC =_FO 0.125_MC ,_, 0.25_MC ,_, 0.375_MC ,_, 0.625_MC and_CC hence_RR &lsqb;_( formula_NN1 &rsqb;_) may_VM be_VBI taken_VVN as_CSA 1,2,3,5_MC ._. 
Using_VVG (_( 21_MC )_) we_PPIS2 now_RT evaluate_VV0 &lsqb;_( formula_NN1 &rsqb;_) and_CC it_PPH1 may_VM be_VBI checked_VVN that_DD1 postmultiplication_NN1 by_II &lsqb;_( formula_NN1 &rsqb;_) gives_VVZ a_AT1 null_JJ product_NN1 ._. 
We_PPIS2 use_VV0 (_( 22_MC )_) as_CSA in_II (_( 2.6.4_MC )_) to_TO find_VVI &lsqb;_( formula_NN1 &rsqb;_) ,_, we_PPIS2 leave_VV0 the_AT solution_NN1 to_II the_AT reader_NN1 :_: it_PPH1 is_VBZ found_VVN that_CST &lsqb;_( formula_NN1 &rsqb;_) which_DDQ when_CS used_VVN to_II premultiply_RR (_( 22_MC )_) ,_, annihilates_VVZ it_PPH1 ._. 
We_PPIS2 now_RT proceed_VV0 to_TO find_VVI the_AT auxiliary_JJ vectors_NN2 ._. 
If_CS we_PPIS2 write_VV0 &lsqb;_( formula_NN1 &rsqb;_) (_( the_AT first_MD element_NN1 of_IO &lsqb;_( formula_NN1 &rsqb;_) is_VBZ nonzero_NN1 ,_, so_CS21 that_CS22 of_IO &lsqb;_( formula_NN1 &rsqb;_) may_VM be_VBI taken_VVN as_CSA zero_MC )_) then_RT we_PPIS2 solve_VV0 ,_, using_VVG the_AT appropriate_JJ submatrices_NN2 (_( see(19)_FO )_) &lsqb;_( formula_NN1 &rsqb;_) The_AT solution_NN1 is_VBZ &lsqb;_( formula_NN1 &rsqb;_) ._. 
A_AT1 similar_JJ calculation_NN1 yields_NN2 &lsqb;_( formula_NN1 &rsqb;_) ._. 
We_PPIS2 must_VM now_RT normalise_VVI these_DD2 ._. 
We_PPIS2 evaluate_VV0 &lsqb;_( formula_NN1 &rsqb;_) Note_VV0 that_CST here_RL ,_, since_CS the_AT second_MD row_NN1 and_CC first_MD column_NN1 of_IO the_AT product_NN1 are_VBR necessarily_RR orthogonal_JJ ,_, m_ZZ1 is_VBZ triangular_JJ ;_; also_RR in_II this_DD1 case_NN1 it_PPH1 happens_VVZ that_CST m-1_FO =_FO m_ZZ1 ._. 
Accordingly_RR ,_, we_PPIS2 may_VM choose_VVI to_TO retain_VVI the_AT columns_NN2 &lsqb;_( formula_NN1 &rsqb;_) and_CC to_TO adopt_VVI the_AT rows_NN2 ,_, orthogonal_JJ to_II &lsqb;_( formula_NN1 &rsqb;_) ,_, &lsqb;_( formula_NN1 &rsqb;_) ,_, &lsqb;_( formula_NN1 &rsqb;_) We_PPIS2 are_VBR now_RT able_JK to_TO evaluate_VVI &lsqb;_( formula_NN1 &rsqb;_) and_CC &lsqb;_( formula_NN1 &rsqb;_) We_PPIS2 can_VM now_RT deflate_VVI A_ZZ1 (_( see_VV0 (_( 21_MC )_) )_) to_II &lsqb;_( formula_NN1 &rsqb;_) Note_VV0 that_CST trA1_FO =_FO 7.5_MC ;_; also_RR A1_FO is_VBZ doubly_RR degenerate_JJ ,_, being_VBG annihilated_VVN by_II postmultiplication_NN1 by_II both_RR &lsqb;_( formula_NN1 &rsqb;_) and_CC &lsqb;_( formula_NN1 &rsqb;_) ._. 
Iteration_NN1 with_IW A1_FO on_II co_NN1 =_FO e4_FO produces_VVZ quite_RG quickly_RR the_AT simple_JJ eigenvalue_NN1 &lsqb;_( formula_NN1 &rsqb;_) 3_MC =_FO 5_MC and_CC vector_NN1 x3_FO =_FO 1,0_MC ,_, -1,1_MC ._. 
We_PPIS2 then_RT evaluate_VV0 5I_FO A1_FO and_CC employ_VV0 (_( 2.6.4_MC )_) to_TO obtain_VVI &lsqb;_( formula_NN1 &rsqb;_) ._. 
Normalising_VVG this_DD1 with_IW x3_FO ,_, we_PPIS2 obtain_VV0 &lsqb;_( formula_NN1 &rsqb;_) and_CC we_PPIS2 are_VBR now_RT able_JK to_TO deflate_VVI A_ZZ1 further_RRR to_II &lsqb;_( formula_NN1 &rsqb;_) Since_CS &lsqb;_( formula_NN1 &rsqb;_) and_CC &lsqb;_( formula_NN1 &rsqb;_) is_VBZ the_AT only_JJ remaining_JJ eigenvalue_NN1 ,_, &lsqb;_( formula_NN1 &rsqb;_) and_CC &lsqb;_( formula_NN1 &rsqb;_) can_VM be_VBI written_VVN &lsqb;_( formula_NN1 &rsqb;_) which_DDQ completes_VVZ the_AT solution_NN1 ._. 
The_AT results_NN2 may_VM be_VBI expressed_VVN compendiously_RR as_CSA &lsqb;_( formula_NN1 &rsqb;_) The_AT numbers_NN2 are_VBR not_XX of_RR21 course_RR22 unique_JJ ._. 
We_PPIS2 could_VM ,_, for_REX21 example_REX22 ,_, divide_VV0 x4_FO by_II any_DD factor_NN1 f_ZZ1 provided_CS we_PPIS2 multiply_VV0 &lsqb;_( formula_NN1 &rsqb;_) by_II f_ZZ1 ._. 
However_RR ,_, if_CS for_REX21 example_REX22 we_PPIS2 divide_VV0 &lsqb;_( formula_NN1 &rsqb;_) by_II 10_MC and_CC multiply_VV0 &lsqb;_( formula_NN1 &rsqb;_) by_II 10_MC ,_, we_PPIS2 must_VM replace_VVI the_AT unit_NN1 in_II the_AT superdiagonal_JJ of_IO &lsqb;_( formula_NN1 &rsqb;_) by_II 10._MC 2.7.6_MC Conjugate_NP1 complex_JJ eigenvalues_NN2 and_CC vectors_NN2 of_IO a_AT1 real_JJ matrix_NN1 So_RG far_RR ,_, we_PPIS2 have_VH0 restricted_VVN our_APPGE studies_NN2 to_II real_JJ eigenvalues_NN2 ._. 
However_RR ,_, complex_JJ eigenvalues_NN2 and_CC vectors_NN2 are_VBR of_IO frequent_JJ occurrence_NN1 ,_, and_CC require_VV0 special_JJ consideration_NN1 ._. 
First_MD ,_, if_CS the_AT matrix_NN1 is_VBZ real_JJ ,_, then_RT its_APPGE characteristic_JJ equation_NN1 must_VM have_VHI real_JJ coefficients_NN2 ._. 
Hence_RR ,_, if_CS it_PPH1 is_VBZ satisfied_VVN by_II &lsqb;_( formula_NN1 &rsqb;_) ,_, then_RT it_PPH1 will_VM also_RR be_VBI satisfied_VVN by_II &lsqb;_( formula_NN1 &rsqb;_) ;_; complex_JJ eigenvalues_NN2 of_IO a_AT1 real_JJ matrix_NN1 therefore_RR occur_VV0 in_II conjugate_NN1 pairs_NN2 ._. 
In_II a_AT1 similar_JJ way_NN1 ,_, if_CS the_AT eigenvector_NN1 corresponding_VVG to_II &lsqb;_( formula_NN1 &rsqb;_) is_VBZ &lsqb;_( formula_NN1 &rsqb;_) ,_, then_RT the_AT relation_NN1 &lsqb;_( formula_NN1 &rsqb;_) implies_VVZ the_AT conjugate_NN1 relation_NN1 ;_; hence_RR p_NNU iq_NN1 is_VBZ the_AT eigenvector_NN1 corresponding_VVG to_II the_AT eigenvalue_NN1 &lsqb;_( formula_NN1 &rsqb;_) Evidently_RR ,_, the_AT vector_NN1 in_II (_( 1_MC1 )_) is_VBZ arbitrary_JJ to_II a_AT1 scalar_JJ complex_JJ multiplier_NN1 ,_, say_VV0 &lsqb;_( formula_NN1 &rsqb;_) +_FO &lsqb;_( formula_NN1 &rsqb;_) another_DD1 vector_NN1 satisfying_JJ (_( 1_MC1 )_) is_VBZ thus_RR &lsqb;_( formula_NN1 &rsqb;_) Any_DD arbitrary_JJ linear_JJ combination_NN1 of_IO p_ZZ1 and_CC q_ZZ1 may_VM therefore_RR be_VBI taken_VVN as_II the_AT real_JJ part_NN1 of_IO the_AT vector_NN1 ,_, with_IW a_AT1 corresponding_JJ combination_NN1 for_IF the_AT imaginary_JJ part_NN1 ._. 
If_CS we_PPIS2 expand_VV0 (_( 1_MC1 )_) we_PPIS2 obtain_VV0 successively_RR &lsqb;_( formula_NN1 &rsqb;_) If_CS we_PPIS2 eliminate_VV0 q_ZZ1 we_PPIS2 find_VV0 &lsqb;_( formula_NN1 &rsqb;_) The_AT same_DA equation_NN1 is_VBZ satisfied_VVN by_II q_ZZ1 ._. 
This_DD1 suggests_VVZ that_CST the_AT square_JJ matrix_NN1 in_II it_PPH1 is_VBZ doubly_RR degenerate_JJ and_CC that_CST p_ZZ1 ,_, q_ZZ1 can_VM each_DD1 have_VHI two_MC arbitrary_JJ elements_NN2 ;_; but_CCB this_DD1 is_VBZ not_XX so_RR ._. 
When_RRQ &lsqb;_( formula_NN1 &rsqb;_) p_ZZ1 are_VBR known_VVN ,_, q_ZZ1 is_VBZ uniquely_RR related_VVN to_II p_ZZ1 by_II &lsqb;_( formula_NN1 &rsqb;_) in_II which_DDQ &lsqb;_( formula_NN1 &rsqb;_) is_VBZ not_XX singular_JJ ._. 
What_DDQ is_VBZ said_VVN above_RL gives_VVZ a_AT1 background_NN1 to_II what_DDQ follows_VVZ ._. 
We_PPIS2 assume_VV0 once_RR21 more_RR22 than_CSN &lsqb;_( formula_NN1 &rsqb;_) have_VH0 dominant_JJ moduli_NN2 ,_, any_DD eigenvalues_NN2 of_IO greater_JJR moduli_NN2 having_VHG been_VBN removed_VVN by_II deflation_NN1 etc_RA ._. 
Again_RT ,_, a_JJ21 priori_JJ22 ,_, we_PPIS2 assume_VV0 nothing_PN1 is_VBZ known_VVN of_IO the_AT nature_NN1 of_IO A_ZZ1 ,_, so_CS21 that_CS22 the_AT process_NN1 of_IO finding_VVG the_AT dominant_JJ eigenvalue_NN1 and_CC its_APPGE vector_NN1 is_VBZ approached_VVN as_RR21 usual_RR22 by_II iterating_VVG with_IW A_ZZ1 on_II an_AT1 arbitrary_JJ column_NN1 co_NN1 ._. 
Once_RR21 again_RR22 ,_, for_IF simplicity_NN1 ,_, we_PPIS2 assume_VV0 A_ZZ1 to_TO be_VBI of_IO order_NN1 4_MC ._. 
The_AT Sylvester_NP1 expansion_NN1 of_IO A_ZZ1 gives_VVZ &lsqb;_( formula_NN1 &rsqb;_) Accordingly_RR ,_, if_CS we_PPIS2 perform_VV0 the_AT usual_JJ iteration_NN1 on_II co_NN1 and_CC continue_VV0 until_CS ,_, with_IW r_ZZ1 =_FO s_ZZ1 ,_, terms_NN2 in_II &lsqb;_( formula_NN1 &rsqb;_) 3_MC and_CC &lsqb;_( formula_NN1 &rsqb;_) 4_MC are_VBR negligible._NNU &lsqb;_( formula_NN1 &rsqb;_) where_RRQ &lsqb;_( formula_NN1 &rsqb;_) is_VBZ a_AT1 complex_JJ scalar_JJ ._. 
Now_RT ,_, as_CSA we_PPIS2 have_VH0 seen_VVN (_( p_ZZ1 +_FO iq_NN1 )_) is_VBZ arbitrary_JJ to_II a_AT1 complex_JJ scalar_JJ multiplier_NN1 (_( and_CC p_ZZ1 iq_VV0 similarly_RR )_) ._. 
We_PPIS2 are_VBR therefore_RR at_II liberty_NN1 to_TO choose_VVI &lsqb;_( formula_NN1 &rsqb;_) and_CC then_RT &lsqb;_( formula_NN1 &rsqb;_) Elimination_NN1 of_IO p_ZZ1 ,_, 1_MC1 from_II these_DD2 three_MC equations_NN2 yields_NN2 &lsqb;_( formula_NN1 &rsqb;_) Note_VV0 that_CST ,_, since_CS &lsqb;_( formula_NN1 &rsqb;_) this_DD1 is_VBZ precisely_RR the_AT same_DA as_CSA Equation_NN1 (_( 3_MC )_) ;_; however_RR ,_, in_II the_AT iteration_NN1 case_NN1 ,_, it_PPH1 is_VBZ only_RR true_JJ when_CS the_AT iterations_NN2 have_VH0 proceeded_VVN sufficiently_RR far_JJ for_IF the_AT terms_NN2 in_II &lsqb;_( formula_NN1 &rsqb;_) 3_MC ,_, &lsqb;_( formula_NN1 &rsqb;_) 4_MC to_TO disappear_VVI ,_, when_CS cs_FO consists_VVZ only_RR of_IO p_ZZ1 ._. 
Now_RT in_II practice_NN1 ,_, with_IW a_AT1 pair_NN of_IO complex_JJ eigenvalues_NN2 ,_, the_AT elements_NN2 in_II &lsqb;_( formula_NN1 &rsqb;_) tend_VV0 to_TO change_VVI sign_NN1 periodically_RR :_: there_EX is_VBZ no_AT point_NN1 in_II reducing_VVG a_AT1 homologous_JJ element_NN1 to_II unity_NN1 ,_, since_CS there_EX is_VBZ no_AT convergence_NN1 to_II a_AT1 settled_JJ form_NN1 ._. 
When_RRQ &lsqb;_( formula_NN1 &rsqb;_) behave_VV0 in_II this_DD1 way_NN1 ,_, conjugate_VV0 complex_JJ eigenvalues_NN2 must_VM be_VBI suspected_VVN ;_; and_CC then_RT three_MC consecutive_JJ columns_NN2 must_VM be_VBI periodically_RR tested_VVN to_TO see_VVI whether_CSW various_JJ pairs_NN2 of_IO homologous_JJ elements_NN2 give_VV0 consistent_JJ values_NN2 for_IF &lsqb;_( formula_NN1 &rsqb;_) ._. 
When_CS this_DD1 happens_VVZ ,_, the_AT iteration_NN1 has_VHZ gone_VVN far_RR enough_RR ,_, and_CC we_PPIS2 can_VM take_VVI p_ZZ1 =_FO cs_FO ,_, when_CS q_ZZ1 is_VBZ determined_VVN from_II the_AT second_MD of_IO equations_NN2 (_( 6_MC )_) ,_, which_DDQ is_VBZ in_II fact_NN1 the_AT same_DA as_CSA (_( 4_MC )_) ._. 
We_PPIS2 now_RT know_VV0 gm_NNU ,_, go_VV0 p_ZZ1 and_CC q_ZZ1 ._. 
Finding_NN1 &lsqb;_( formula_NN1 &rsqb;_) is_VBZ not_XX so_RG straightforward_JJ ._. 
Knowing_VVG &lsqb;_( formula_NN1 &rsqb;_) we_PPIS2 can_VM either(i)Solve_VVI &lsqb;_( formula_NN1 &rsqb;_) with_IW one_MC1 element_NN1 of_IO &lsqb;_( formula_NN1 &rsqb;_) arbitrarily_RR assigned_VVN ._. 
We_PPIS2 then_RT have_VH0 to_TO solve_VVI a_AT1 set_NN1 of_IO linear_JJ algebraic_JJ equations_NN2 with_IW complex_JJ coefficients_NN2 ;_; (_( ii_MC )_) Iterate_VV0 with_IW A_ZZ1 postmultiplying_VVG an_AT1 arbitrary_JJ row_NN1 &lsqb;_( formula_NN1 &rsqb;_) ;_; this_DD1 will_VM lead_VVI to_II &lsqb;_( formula_NN1 &rsqb;_) (_( and_CC &lsqb;_( formula_NN1 &rsqb;_) again_RT )_) as_II21 for_II22 p_ZZ1 ,_, q._NNU (_( iii_MC )_) Solve_VV0 the_AT equations_NN2 which_DDQ parallel_NN1 (_( 3_MC )_) and(4)_FO :_: &lsqb;_( formula_NN1 &rsqb;_) Since_CS &lsqb;_( formula_NN1 &rsqb;_) are_VBR known_VVN ,_, this_DD1 ,_, like_II (_( ii_MC )_) ,_, involves_VVZ only_RR real_JJ numbers_NN2 ;_; on_II the_AT other_JJ hand_NN1 ,_, it_PPH1 requires_VVZ &lsqb;_( formula_NN1 &rsqb;_) ,_, the_AT evaluation_NN1 of_IO which_DDQ requires_VVZ considerable_JJ computer_NN1 time_NNT1 if_CS the_AT order_NN1 of_IO A_ZZ1 is_VBZ large_JJ ._. 
On_II the_AT whole_NN1 ,_, since_CS most_DAT modern_JJ computers_NN2 can_VM work_VVI with_IW complex_JJ numbers_NN2 ,_, (_( i_ZZ1 )_) seems_VVZ the_AT most_RGT straightforward_JJ ._. 
When_CS we_PPIS2 have_VH0 obtained_VVN &lsqb;_( formula_NN1 &rsqb;_) we_PPIS2 can_VM perform_VVI a_AT1 check_NN1 ,_, which_DDQ is_VBZ in_II any_DD event_NN1 needed_VVN when_CS the_AT vectors_NN2 are_VBR normalised_VVN ._. 
For_IF since_RR &lsqb;_( formula_NN1 &rsqb;_) belong_VV0 to_II different_JJ eigenvalues_NN2 ,_, they_PPHS2 are_VBR orthogonal_JJ ,_, and_CC &lsqb;_( formula_NN1 &rsqb;_) so_CS21 that_CS22 &lsqb;_( formula_NN1 &rsqb;_) For_IF the_AT process_NN1 of_IO normalisation_NN1 ,_, we_PPIS2 require_VV0 to_TO evaluate_VVI &lsqb;_( formula_NN1 &rsqb;_) and_CC m_ZZ1 is_VBZ evidently_RR diagonal_JJ ;_; in_II31 view_II32 of_II33 the_AT check_NN1 equations_NN2 (_( 9_MC )_) it_PPH1 can_VM be_VBI written_VVN &lsqb;_( formula_NN1 &rsqb;_) the_AT determinant_NN1 of_IO which_DDQ is_VBZ &lsqb;_( formula_NN1 &rsqb;_) ,_, so_CS its_APPGE reciprocal_JJ is_VBZ readily_RR written_VVN down_RP and_CC we_PPIS2 can_VM obtain_VVI normalised_JJ forms_NN2 of_IO &lsqb;_( formula_NN1 &rsqb;_) ._. 
We_PPIS2 can_VM now_RT deflate_VVI A._NNU Since_CS &lsqb;_( formula_NN1 &rsqb;_) a_AT1 deflated_JJ matrix_NN1 A1_FO is_VBZ &lsqb;_( formula_NN1 &rsqb;_) We_PPIS2 can_VM therefore_RR use_VVI A1_FO to_TO obtain_VVI the_AT remaining_JJ eigenvalues_NN2 and_CC vectors_NN2 of_IO A_ZZ1 in_II the_AT usual_JJ way_NN1 ._. 
Example_NN1 1_MC1 Let_VV0 &lsqb;_( formula_NN1 &rsqb;_) If_CS we_PPIS2 use_VV0 this_DD1 for_IF iterative_JJ premultiplication_NN1 of_IO an_AT1 arbitrary_JJ column_NN1 ,_, say_VV0 the_AT summing_JJ vector_NN1 ,_, successive_JJ columns_NN2 are_VBR as_CSA show_NN1 in_II Table_NN1 1._MC &lsqb;_( formula_NN1 &rsqb;_) It_PPH1 is_VBZ apparent_JJ from_II the_AT first_MD that_CST the_AT vector_NN1 elements_NN2 are_VBR changing_JJ sign_NN1 ;_; so_RR we_PPIS2 do_VD0 not_XX reduce_VVI a_AT1 homologous_JJ element_NN1 to_II unity_NN1 (_( though_CS in_RR21 general_RR22 we_PPIS2 might_VM periodically_RR remove_VVI a_AT1 power_NN1 of_IO 10_MC )_) ._. 
If_CS ,_, starting_VVG say_VV0 at_II c5_FO ,_, we_PPIS2 test_VV0 to_TO see_VVI if_CSW pairs_NN2 of_IO homologous_JJ elements_NN2 give_VV0 consistent_JJ values_NN2 for_IF &lsqb;_( formula_NN1 &rsqb;_) ,_, &lsqb;_( formula_NN1 &rsqb;_) in_II (_( 7_MC )_) ,_, we_PPIS2 find_VV0 this_DD1 is_VBZ not_XX so_RR ._. 
However_RR ,_, &lsqb;_( formula_NN1 &rsqb;_) give_VV0 nearly_RR consistent_JJ values_NN2 ;_; we_PPIS2 therefore_RR begin_VV0 with_IW &lsqb;_( formula_NN1 &rsqb;_) accurately_RR :_: &lsqb;_( formula_NN1 &rsqb;_) If_CS these_DD2 three_MC columns_NN2 are_VBR used_VVN in_II (_( 7_MC )_) then_RT any_DD two_MC pairs_NN2 of_IO homologous_JJ elements_NN2 give_VV0 &lsqb;_( formula_NN1 &rsqb;_) ._. 
We_PPIS2 thus_RR obtain_VV0 &lsqb;_( formula_NN1 &rsqb;_) Also_RR ,_, we_PPIS2 may_VM now_RT choose_VVI &lsqb;_( formula_NN1 &rsqb;_) and_CC using_VVG the_AT second_MD of_IO Equations_NN2 (_( 6_MC )_) we_PPIS2 now_RT find_VV0 &lsqb;_( formula_NN1 &rsqb;_) We_PPIS2 now_RT have_VH0 to_TO find_VVI &lsqb;_( formula_NN1 &rsqb;_) ,_, and_CC for_IF this_DD1 we_PPIS2 use_VV0 (_( 8_MC )_) ._. 
One_MC1 element_NN1 of_IO &lsqb;_( formula_NN1 &rsqb;_) can_VM be_VBI arbitrarily_RR assigned_VVN :_: here_RL we_PPIS2 assign_VV0 the_AT value_NN1 1_MC1 to_II the_AT third_MD element_NN1 and_CC solve_VV0 &lsqb;_( formula_NN1 &rsqb;_) which_DDQ ,_, if_CS we_PPIS2 omit_VV0 one_MC1 column_NN1 (_( e.g._REX the_AT last_MD )_) from_II the_AT square_JJ matrix_NN1 ,_, gives_VVZ &lsqb;_( formula_NN1 &rsqb;_) We_PPIS2 omit_VV0 the_AT details_NN2 of_IO solution_NN1 ;_; the_AT answer_NN1 is_VBZ &lsqb;_( formula_NN1 &rsqb;_) Note_VV0 that_CST &lsqb;_( formula_NN1 &rsqb;_) Accordingly_RR ,_, &lsqb;_( formula_NN1 &rsqb;_) and_CC if_CS we_PPIS2 postmultiply_RR p_ZZ1 +_FO iq_VV0 as_CSA given_VVN by_II (_( 14_MC )_) and_CC (_( 15_MC )_) my_APPGE m-1_FO we_PPIS2 obtain_VV0 as_CSA our_APPGE normalised_JJ p_ZZ1 ,_, q_ZZ1 ,_, &lsqb;_( formula_NN1 &rsqb;_) We_PPIS2 are_VBR now_RT able_JK to_TO deflate_VVI A._NNU Equation_NN1 (_( 11_MC )_) becomes_VVZ ,_, with_IW A_ZZ1 given_VVN by_II (_( 12_MC )_) and_CC use_NN1 of_IO (_( 13_MC )_) ,_, (_( 14_MC )_) ,_, (_( 15_MC )_) and_CC (_( 16_MC )_) ,_, &lsqb;_( formula_NN1 &rsqb;_) A1_FO is_VBZ ,_, in_II fact_NN1 ,_, doubly_RR degenerate_JJ ;_; it_PPH1 is_VBZ annihilated_VVN by_II both_DB2 the_AT columns_NN2 immediately_RR above_II it._NNU iteration_NN1 with_IW A1_FO on_II an_AT1 arbitrary_JJ column_NN1 quickly_RR leads_VVZ to_II the_AT result_NN1 &lsqb;_( formula_NN1 &rsqb;_) Further_JJR delation_NN1 leads_VVZ at_RR21 once_RR22 to_II &lsqb;_( formula_NN1 &rsqb;_) which_DDQ completes_VVZ the_AT solution._NNU 2.7.7_MC Adjacent_JJ eigenvalues_NN2 :_: Two_MC eigenvalues_NN2 with_IW nearly-equal_JJ moduli_NN2 When_RRQ two_MC eigenvalues_NN2 have_VH0 nearly-equal_JJ moduli_NN2 ,_, it_PPH1 is_VBZ obvious_JJ that_CST very_RG many_DA2 iterations_NN2 will_VM be_VBI required_VVN for_IF a_AT1 direct_JJ solution_NN1 ._. 
The_AT following_JJ device_NN1 can_VM be_VBI helpful_JJ in_II this_DD1 case_NN1 ._. 
Suppose_VV0 the_AT iterations_NN2 have_VH0 proceeded_VVN so_RG far_RR that_DD1 only_JJ contributions_NN2 from_II the_AT two_MC nearly-equal_JJ eigenvalues_NN2 remain_VV0 ._. 
Then_RT we_PPIS2 can_VM write_VVI ,_, when_CS a_AT1 homologous_JJ element_NN1 is_VBZ not_XX made_VVN unity_NN1 ,_, &lsqb;_( formula_NN1 &rsqb;_) Here_RL a_AT1 ,_, b_ZZ1 ,_, c_ZZ1 ,_, d_ZZ1 ;_; p_ZZ1 ,_, q_ZZ1 are_VBR the_AT values_NN2 of_IO a_AT1 homologous_JJ element_NN1 in_II the_AT various_JJ columns_NN2 ;_; a_AT1 ,_, b_ZZ1 ,_, c_ZZ1 ,_, d_ZZ1 are_VBR known_VVN ._. 
Elimination_NN1 of_IO p_ZZ1 ,_, q_ZZ1 ,_, yields_VVZ two_MC equations_NN2 which_DDQ may_VM be_VBI solved_VVN for_IF &lsqb;_( formula_NN1 &rsqb;_) 1_MC1 +_FO &lsqb;_( formula_NN1 &rsqb;_) 2_MC and_CC &lsqb;_( formula_NN1 &rsqb;_) 1&lsqb;formula&rsqb;2_FO :_: &lsqb;_( formula_NN1 &rsqb;_) Hence_RR ,_, &lsqb;_( formula_NN1 &rsqb;_) ,_, &lsqb;_( formula_NN1 &rsqb;_) are_VBR readily_RR found_VVN ._. 
When_CS they_PPHS2 are_VBR known_VVN ,_, then_RT &lsqb;_( formula_NN1 &rsqb;_) which_DDQ follow_VV0 from_II the_AT first_MD two_MC of_IO Equations_NN2 (_( 1_MC1 )_) ._. 
When_RRQ &lsqb;_( formula_NN1 &rsqb;_) 1_MC1 ,_, &lsqb;_( formula_NN1 &rsqb;_) 2_MC are_VBR very_RG close_JJ ,_, both_RR numerator_NN1 and_CC denominator_NN1 in_II (_( 2_MC )_) tend_VV0 to_TO be_VBI small_JJ differences_NN2 ,_, so_CS21 that_CS22 accurate_JJ calculation_NN1 of_IO a_AT1 ,_, b_ZZ1 ,_, c_ZZ1 ,_, d_ZZ1 is_VBZ desirable_JJ ._. 
In_II practice_NN1 ,_, iteration_NN1 in_II this_DD1 case_NN1 behaves_VVZ rather_RR like_II that_DD1 for_IF a_AT1 defective_JJ matrix_NN1 (_( 2.7.5_MC )_) ,_, in_II that_DD1 convergence_NN1 is_VBZ slow_JJ and_CC successive_JJ columns_NN2 tend_VV0 to_II a_AT1 quasi-arithmetic_NN1 progression_NN1 of_IO the_AT elements_NN2 ._. 
The_AT occurrence_NN1 of_IO nearly-equal_JJ roots_NN2 is_VBZ more_RGR common_JJ than_CSN that_DD1 of_IO a_AT1 defective_JJ matrix_NN1 ;_; hence_RR tests_VVZ to_TO see_VVI whether_CSW homologous_JJ elements_NN2 yield_VV0 ,_, in_II (_( 2_MC )_) ,_, consistent_JJ results_NN2 should_VM be_VBI applied_VVN first_MD when_CS convergence_NN1 is_VBZ slow._NNU &lsqb;_( formula_NN1 &rsqb;_) Example_NN1 1_MC1 &lsqb;_( formula_NN1 &rsqb;_) Iteration_NN1 on_II a_AT1 column_NN1 ,_, beginning_VVG with_IW co_NN1 =_FO e4_FO ,_, yields_VVZ successive_JJ columns_NN2 as_CSA given_VVN in_II Table_NN1 1_MC1 ._. 
At_II this_DD1 stage_NN1 ,_, we_PPIS2 make_VV0 consecutive_JJ accurate_JJ calculations_NN2 ,_, to_TO see_VVI if_CS we_PPIS2 can_VM get_VVI consistent_JJ values_NN2 for_IF &lsqb;_( formula_NN1 &rsqb;_) :_: &lsqb;_( formula_NN1 &rsqb;_) Using_VVG the_AT elements_NN2 of_IO the_AT last_MD line_NN1 in_II (_( 2_MC )_) we_PPIS2 obtain_VV0 &lsqb;_( formula_NN1 &rsqb;_) The_AT elements_NN2 of_IO the_AT first_MD line_NN1 give_VV0 &lsqb;_( formula_NN1 &rsqb;_) The_AT results_NN2 are_VBR consistent_JJ ,_, as_CSA indeed_RR are_VBR the_AT results_NN2 from_II the_AT second_MD and_CC third_MD lines_NN2 ._. 
By_II inspection_NN1 (_( or_CC from_II the_AT implied_JJ quadratic_JJ )_) &lsqb;_( formula_NN1 &rsqb;_) We_PPIS2 now_RT evaluate_VV0 the_AT eigenvectors_NN2 :_: &lsqb;_( formula_NN1 &rsqb;_) so_CS21 that_CS22 we_PPIS2 may_VM take_VVI &lsqb;_( formula_NN1 &rsqb;_) I_PPIS1 now_RT remains_VVZ to_TO find_VVI the_AT corresponding_JJ row_NN1 vectors_NN2 and_CC to_TO deflate_VVI A_ZZ1 for_IF further_JJR study_NN1 ;_; we_PPIS2 leave_VV0 this_DD1 to_II the_AT reader_NN1 ._. 
However_RR ,_, we_PPIS2 may_VM note_VVI here_RL that_CST &lsqb;_( formula_NN1 &rsqb;_) whence_RRQ &lsqb;_( formula_NN1 &rsqb;_) ._. 
2.7.8_MC Applicability_NN1 of_IO power_NN1 methods_NN2 We_PPIS2 have_VH0 seen_VVN that_CST the_AT power_NN1 method_NN1 can_VM be_VBI used_VVN to_TO obtain_VVI dominant_JJ eigenvalues_NN2 and_CC the_AT associated_JJ vectors_NN2 ,_, and_CC ,_, by_II deflation_NN1 ,_, ,_, shifting_VVG and_CC inverse_JJ iteration_NN1 ,_, can_VM also_RR be_VBI used_VVN to_TO find_VVI all_DB non-dominant_JJ eigenvalues_NN2 and_CC their_APPGE vectors_NN2 ;_; the_AT results_NN2 are_VBR obtained_VVN seriatim_NN1 ._. 
This_DD1 implies_VVZ that_CST ,_, if_CS one_PN1 is_VBZ interested_JJ only_RR in_II a_AT1 limited_JJ number_NN1 of_IO eigenvalues_NN2 ,_, the_AT power_NN1 method_NN1 is_VBZ the_AT obvious_JJ choice_NN1 ._. 
For_REX21 example_REX22 ,_, suppose_VV0 we_PPIS2 have_VH0 a_AT1 mathematical_JJ model_NN1 with_IW n_ZZ1 =_FO 100_MC for_IF study_NN1 of_IO the_AT vibration_NN1 characteristics_NN2 of_IO an_AT1 aircraft_NN ._. 
The_AT aeroelastic_JJ engineer_NN1 will_VM usually_RR be_VBI interested_JJ only_RR in_II the_AT fundamental_JJ mode_NN1 of_IO vibration_NN1 and_CC a_AT1 few_DA2 overtones_NN2 ,_, say_VV0 10_MC in_II all_DB ;_; in_II the_AT usual_JJ model_NN1 this_DD1 implies_VVZ the_AT dominant_JJ and_CC the_AT 9_MC immediately_RR subdominant_JJ eigenvalues_NN2 and_CC vectors_NN2 ._. 
The_AT vibration_NN1 engineer_NN1 will_VM be_VBI concerned_JJ with_IW a_AT1 band_NN1 of_IO frequencies_NN2 near_II the_AT engine_NN1 rotational_JJ speed_NN1 ,_, and_CC so_RR will_VM use_VVI shifting_JJ and_CC inverse_JJ iteration_NN1 ._. 
But_CCB in_II any_DD event_NN1 ,_, the_AT mathematical_JJ model_NN1 will_VM usually_RR involve_VVI the_AT finite_JJ element_NN1 concept_NN1 which_DDQ gives_VVZ accurate_JJ values_NN2 for_IF the_AT lower_JJR band_NN1 of_IO frequencies_NN2 ,_, but_CCB is_VBZ often_RR quite_RG unrepresentative_JJ of_IO the_AT top_JJ frequencies_NN2 ._. 
Evaluation_NN1 of_IO all_DB frequencies_NN2 and_CC modes_NN2 is_VBZ therefore_RR not_XX normally_RR required_VVN ._. 
For_IF such_DA a_AT1 model_NN1 ,_, power_NN1 methods_NN2 are_VBR the_AT obvious_JJ choice_NN1 ._. 
However_RR ,_, if_CS interest_NN1 attaches_VVZ to_II all_DB eigenvalues_NN2 ,_, and_CC in_RR21 particular_RR22 if_CS there_EX is_VBZ less_DAR concern_NN1 with_IW the_AT associated_JJ vectors_NN2 ,_, other_JJ methods_NN2 may_VM be_VBI used_VVN ._. 
We_PPIS2 now_RT begin_VV0 a_AT1 brief_JJ study_NN1 of_IO transformation_NN1 methods._NNU 2.8_MC TRANSFORMATION_NN1 METHODS_NN2 In_II these_DD2 methods_NN2 ,_, we_PPIS2 apply_VV0 a_AT1 succession_NN1 of_IO similar_JJ transformations_NN2 to_II a_AT1 matrix_NN1 A_ZZ1 :_: &lsqb;_( formula_NN1 &rsqb;_) and_RR31 so_RR32 on_RR33 ,_, until_CS A_ZZ1 is_VBZ transformed_VVN into_II a_AT1 form_NN1 giving_VVG the_AT eigenvalues_NN2 directly_RR :_: it_PPH1 may_VM be_VBI a_AT1 triangular_JJ form_NN1 ,_, or_CC the_AT ultimate_JJ canonical_JJ form_NN1 &lsqb;_( formula_NN1 &rsqb;_) (_( or_CC &lsqb;_( formula_NN1 &rsqb;_) if_CS A_ZZ1 is_VBZ defective_JJ )_) ._. 
Since_CS the_AT transformations_NN2 are_VBR similar_JJ ,_, all_DB the_AT derived_JJ matrices_NN2 B_ZZ1 ,_, C_ZZ1 ,_, D_ZZ1 ,_, ..._... have_VH0 the_AT same_DA eigenvalues_NN2 as_CSA A_ZZ1 ;_; and_CC if_CS we_PPIS2 proceed_VV0 to_II the_AT diagonal_JJ form_NN1 ,_, then_RT the_AT chain_NN1 product_NN1 &lsqb;_( formula_NN1 &rsqb;_) &lsqb;_( formula_NN1 &rsqb;_) X3_FO ..._... gives_VVZ the_AT modal_JJ matrix_NN1 X._NP1 2.8.1_MC Jacobi_NP1 's_GE method_NN1 for_IF a_AT1 real_JJ symmetric_JJ matrix_NN1 In_II 1.22_MC ,_, Theorems_NN2 VIII_MC and_CC X_ZZ1 ,_, we_PPIS2 have_VH0 shown_VVN that_CST a_AT1 real_JJ symmetric_JJ matrix_NN1 has_VHZ real_JJ eigenvalues_NN2 and_CC that_CST it_PPH1 can_VM not_XX be_VBI defective_JJ ._. 
Moreover_RR ,_, its_APPGE modal_JJ matrix_NN1 is_VBZ orthogonal_JJ ;_; hence_RR &lsqb;_( formula_NN1 &rsqb;_) Jacobi_NP1 's_GE method_NN1 is_VBZ as_CSA follows_VVZ ._. 
We_PPIS2 begin_VV0 by_II searching_VVG the_AT off-diagonal_JJ elements_NN2 of_IO A_ZZ1 (_( since_CS A_ZZ1 is_VBZ symmetrical_JJ ,_, we_PPIS2 usually_RR use_VV0 the_AT upper_JJ half_NN1 only_RR )_) to_TO find_VVI the_AT element_NN1 Auv_NN1 of_IO greatest_JJT modulus_NN1 ._. 
We_PPIS2 then_RT construct_VV0 the_AT orthogonal_JJ matrix_NN1 &lsqb;_( formula_NN1 &rsqb;_) ,_, which_DDQ is_VBZ the_AT unit_NN1 matrix_NN1 except_CS21 that_CS22 the_AT units_NN2 in_II the_AT (_( u_ZZ1 ,_, u_ZZ1 )_) and_CC (_( v_ZZ1 ,_, v_ZZ1 )_) positions_NN2 are_VBR each_DD1 replaced_VVN by_II &lsqb;_( formula_NN1 &rsqb;_) and_CC the_AT zeros_MC2 in_II the_AT (_( u_ZZ1 ,_, v_ZZ1 )_) and_CC (_( v_ZZ1 ,_, u_ZZ1 )_) positions_NN2 by_II &lsqb;_( formula_NN1 &rsqb;_) and_CC s_ZZ1 respectively_RR ._. 
Then_RT &lsqb;_( formula_NN1 &rsqb;_) and_CC we_PPIS2 may_VM evaluate_VVI &lsqb;_( formula_NN1 &rsqb;_) of_IO which_DDQ the_AT relevant_JJ submatrix_NN1 is_VBZ (_( omitting_VVG all_DB other_JJ rows_NN2 and_CC columns_NN2 )_) &lsqb;_( formula_NN1 &rsqb;_) which_DDQ yields_VVZ &lsqb;_( formula_NN1 &rsqb;_) We_PPIS2 now_RT choose_VV0 gth_NNU 1_MC1 such_CS21 that_CS22 Buv_NP1 vanishes_VVZ ;_; i.e._REX &lsqb;_( formula_NN1 &rsqb;_) This_DD1 fixed_JJ gth_NNU 1_MC1 ._. 
If_CS than_CSN 2gth1_FO is_VBZ positive_JJ ,_, we_PPIS2 take_VV0 2gth1_FO to_TO lie_VVI in_II the_AT first_MD quadrant_NN1 ;_; if_CS negative_JJ ,_, in_II the_AT fourth_MD ,_, Then_RT gth_NNU 1_MC1 lies_VVZ between_II &lsqb;_( formula_NN1 &rsqb;_) =_FO c_ZZ1 is_VBZ positive_JJ and_CC &lsqb;_( formula_NN1 &rsqb;_) has_VHZ the_AT sign_NN1 of_IO tan_NN1 2gth1_FO ._. 
With_IW c_ZZ1 ,_, s_ZZ1 fixed_JJ ,_, we_PPIS2 can_VM now_RT evaluate_VVI B_ZZ1 fully_RR ._. 
Elements_NN2 other_II21 than_II22 in_II the_AT uth_NN1 ,_, vth_NNU columns_NN2 and_CC rows_NN2 are_VBR unaltered_JJ from_II those_DD2 in_II A_ZZ1 ;_; at_II the_AT intersections_NN2 of_IO the_AT uth_NN1 ,_, vth_NNU columns_NN2 an_AT1 rows_NN2 the_AT new_JJ elements_NN2 are_VBR given_VVN by_II (_( 3_MC )_) (_( with_IW Buv_NP1 =_FO 0_MC )_) ,_, while_CS the_AT remaining_JJ elements_NN2 Biu_NN1 ,_, Biv_NP1 in_II the_AT uth_NN1 ,_, vth_NNU columns_NN2 and_CC rows_NN2 are_VBR &lsqb;_( formula_NN1 &rsqb;_) We_PPIS2 may_VM make_VVI two_MC deductions_NN2 ._. 
First_MD ,_, from_II (_( 3_MC )_) we_PPIS2 have_VH0 &lsqb;_( formula_NN1 &rsqb;_) and_CC since_CS only_RR these_DD2 diagonal_JJ elements_NN2 have_VH0 changed_VVN ,_, it_PPH1 follows_VVZ that_CST &lsqb;_( formula_NN1 &rsqb;_) as_CSA it_PPH1 must_VM in_II a_AT1 similar_JJ transformation_NN1 ._. 
Next_MD ,_, fro(5)_FO ,_, squaring_VVG and_CC adding_VVG ,_, we_PPIS2 have_VH0 &lsqb;_( formula_NN1 &rsqb;_) and_CC since_CS only_RR the_AT uth_NN1 ,_, vth_NNU columns_NN2 and_CC rows_NN2 have_VH0 changed_VVN ,_, we_PPIS2 deduce_VV0 that_CST the_AT sum_NN1 of_IO the_AT squares_NN2 of_IO all_DB the_AT off-diagonal_JJ elements_NN2 of_IO B_ZZ1 is_VBZ less_DAR than_CSN that_DD1 of_IO A_ZZ1 since_CS we_PPIS2 have_VH0 replaced_VVN the_AT two_MC elements_NN2 Auv_NN1 of_IO A_ZZ1 by_II zeros_NN2 in_II B._NP1 Thus_RR &lsqb;_( formula_NN1 &rsqb;_) Having_VHG evaluated_VVN B_ZZ1 ,_, we_PPIS2 repeat_VV0 the_AT procedure_NN1 to_TO find_VVI C_NP1 ;_; we_PPIS2 select_VV0 the_AT element_NN1 Bpq_NP1 of_IO largest_JJT modulus_NN1 and_CC find_VV0 &lsqb;_( formula_NN1 &rsqb;_) from_II (_( see_VV0 4_MC )_) )_) &lsqb;_( formula_NN1 &rsqb;_) we_PPIS2 then_RT proceed_VV0 as_CSA before_RT to_TO construct_VVI &lsqb;_( formula_NN1 &rsqb;_) and_CC &lsqb;_( formula_NN1 &rsqb;_) ._. 
C_NP1 will_VM have_VHI Cpq_NP1 =_FO 0_MC and_CC may_VM also_RR have_VHI cuv_NN1 =_FO 0_MC (_( if_CS &lsqb;_( formula_NN1 &rsqb;_) )_) ;_; but_CCB this_DD1 situation_NN1 will_VM not_XX persist_VVI ._. 
At_II some_DD stage_NN1 ,_, in_RR21 general_RR22 ,_, elements_NN2 formerly_RR made_VVN zero_NN1 will_VM again_RT become_VVI nonzero_NN1 ,_, though_CS usually_RR smaller_JJR than_CSN before_RT ._. 
Thus_RR Jacobi_NP1 's_GE method_NN1 does_VDZ not_XX terminate_VVI in_II a_AT1 finite_JJ ,_, predictable_JJ number_NN1 of_IO steps_NN2 ;_; indeed_RR ,_, the_AT umber_NN1 may_VM be_VBI very_RG large_JJ ._. 
Nevertheless_RR ,_, following_RA (_( 6_MC )_) &lsqb;_( formula_NN1 &rsqb;_) and_CC ,_, in_II an_AT1 overall_JJ sense_NN1 ,_, the_AT off-diagonal_JJ elements_NN2 therefore_RR become_VV0 progressively_RR smaller_JJR ,_, until_CS eventually_RR we_PPIS2 obtain_VV0 a_AT1 matrix_NN1 in_II which_DDQ the_AT off-diagonal_JJ elements_NN2 are_VBR all_DB vanishingly_RR small_JJ :_: i.e._REX a_AT1 diagonal_JJ matrix_NN1 of_IO which_DDQ the_AT diagonal_JJ elements_NN2 are_VBR the_AT eigenvalues_NN2 ._. 
We_PPIS2 give_VV0 below_II an_AT1 example_NN1 ,_, but_CCB must_VM enter_VVI the_AT caveat_NN1 that_CST because_II21 of_II22 its_APPGE small_JJ order_NN1 (_( n_ZZ1 =_FO 3_MC )_) it_PPH1 converges_VVZ ra&lsqb;formula&rsqb;dly_RR ._. 
At_II each_DD1 stage_NN1 ,_, the_AT sum_NN1 of_IO the_AT squares_NN2 of_IO the_AT off-diagonal_JJ elements_NN2 decreases_VVZ by_II at_RR21 least_RR22 one_MC1 third_MD ._. 
However_RR ,_, if_CS we_PPIS2 had_VHD a_AT1 matrix_NN1 of_IO order_NN1 100_MC ,_, it_PPH1 would_VM have_VHI 9900_MC off-diagonal_JJ elements_NN2 ,_, and_CC we_PPIS2 can_VM only_RR say_VVI that_CST at_II each_DD1 stage_NN1 the_AT sum_NN1 of_IO the_AT squares_NN2 of_IO these_DD2 elements_NN2 would_VM decrease_VVI by_II 1/4950_MF at_RR21 least_RR22 ;_; convergence_NN1 to_II diagonal_JJ form_NN1 is_VBZ likely_JJ to_TO take_VVI very_RG much_DA1 longer_JJR than_CSN for_IF a_AT1 small_JJ matrix_NN1 ._. 
We_PPIS2 may_VM add_VVI two_MC observations_NN2 ._. 
The_AT method_NN1 is_VBZ viable_JJ if_CS Auv_NP1 is_VBZ reduced_VVN ,_, not_XX to_II zero_MC ,_, but_CCB to_II some_DD lesser_JJ modulus_NN1 ;_; nevertheless_RR ,_, reduction_NN1 to_TO zero_VVI is_VBZ obviously_RR best_JJT ._. 
Also_RR ,_, the_AT method_NN1 requires_VVZ only_RR that_CST A_ZZ1 is_VBZ real_JJ and_CC symmetric_JJ :_: its_APPGE state_NN1 of_IO definiteness_NN1 is_VBZ irrelevant_JJ ._. 
In_II the_AT example_NN1 below_RL ,_, A_ZZ1 has_VHZ a_AT1 negative_JJ eigenvalue_NN1 ._. 
Example_NN1 1_MC1 Let_VV0 &lsqb;_( formula_NN1 &rsqb;_) We_PPIS2 select_VV0 the_AT off-diagonal_JJ element_NN1 of_IO largest_JJT modulus_NN1 (_( underlined_VVN )_) and_CC evaluate_VV0 (_( we_PPIS2 record_VV0 only_RR six_MC decimal_JJ places_NN2 )_) &lsqb;_( formula_NN1 &rsqb;_) Hence_RR &lsqb;_( formula_NN1 &rsqb;_) and_CC then_RT &lsqb;_( formula_NN1 &rsqb;_) Note_VV0 that_CST &lsqb;_( formula_NN1 &rsqb;_) also_RR &lsqb;_( formula_NN1 &rsqb;_) thus_RR we_PPIS2 have_VH0 effected_VVN a_AT1 considerable_JJ overall_JJ reduction_NN1 ._. 
The_AT second_MD step_NN1 is_VBZ to_TO obtain_VVI C._NP1 For_IF this_DD1 we_PPIS2 select_VV0 the_AT underlined_JJ element_NN1 in_II B_ZZ1 an_AT1 evaluate_VV0 in_II succession_NN1 &lsqb;_( formula_NN1 &rsqb;_) and_CC then_RT &lsqb;_( formula_NN1 &rsqb;_) Note_VV0 that_CST the_AT zero_NN1 elements_NN2 in_II B_ZZ1 have_VH0 become_VVN nonzero_NN1 in_II C_NP1 ;_; however_RR ,_, &lsqb;_( formula_NN1 &rsqb;_) is_VBZ now_RT 2.19132_MC ;_; compare_VV0 &lsqb;_( formula_NN1 &rsqb;_) above_RL ._. 
Proceeding_VVG in_II this_DD1 way_NN1 ,_, after_II six_MC transformations_NN2 ,_, we_PPIS2 find_VV0 to_II the_AT order_NN1 of_IO accuracy_NN1 employed_VVN &lsqb;_( formula_NN1 &rsqb;_) 2.8.2_MC The_AT LR_NP1 and_CC QR_NP1 methods_NN2 We_PPIS2 now_RT describe_VV0 briefly_RR two_MC methods_NN2 which_DDQ are_VBR applicable_JJ to_II any_DD square_JJ matrix_NN1 A_ZZ1 ,_, whether_CSW symmetric_JJ or_CC not_XX ,_, real_JJ or_CC complex_JJ ,_, simple_JJ or_CC defective_JJ ,_, non-singular_JJ or_CC singular_NN1 ._. 
Here_RL ,_, we_PPIS2 shall_VM merely_RR state_VVI and_CC illustrate_VVI the_AT methods_NN2 ;_; for_IF proofs_NN2 ,_, readers_NN2 are_VBR referred_VVN to_II Wilkinson_NP1 (_( 6_MC )_) ._. 
We_PPIS2 may_VM note_VVI ,_, however_RR ,_, that_CST while_CS a_AT1 complex_JJ matrix_NN1 A_ZZ1 =_FO B_ZZ1 +_FO iC_JJ (_( B_ZZ1 ,_, C_ZZ1 real_JJ )_) may_VM be_VBI treated_VVN as_II such_DA ,_, using_VVG complex_JJ arithmetic_NN1 ,_, it_PPH1 is_VBZ more_RGR usual_JJ to_TO treat_VVI its_APPGE real_JJ equivalent_NN1 &lsqb;_( formula_NN1 &rsqb;_) which_DDQ (_( see_VV0 Theorem_NN1 XI_NN1 ,_, 1.22_MC )_) yields_VVZ the_AT eigenvalues_NN2 and_CC vectors_NN2 of_IO A_ZZ1 and_CC its_APPGE conjugate._NNU 2.8.3_MC The_AT LR_NP1 algorithm_NN1 It_PPH1 is_VBZ a_AT1 straightforward_JJ process_NN1 ,_, readily_RR programmed_VVN for_IF a_AT1 computer_NN1 ,_, to_TO resolve_VVI a_AT1 matrix_NN1 A_ZZ1 into_II the_AT product_NN1 L1R1_FO where_RRQ L1_FO is_VBZ a_AT1 lower_JJR triangular_JJ matrix_NN1 having_VHG units_NN2 in_II its_APPGE diagonal_JJ ,_, and_CC R1_FO is_VBZ an_AT1 upper_JJ triangular_JJ matrix_NN1 ._. 
For_IF a_AT1 model_NN1 ,_, we_PPIS2 here_RL take_VV0 n_ZZ1 =_FO 4_MC ;_; it_PPH1 is_VBZ ty&lsqb;formula&rsqb;cal_JJ of_IO any_DD order_NN1 ._. 
Let_VV0 &lsqb;_( formula_NN1 &rsqb;_) &lsqb;_( formula_NN1 &rsqb;_) Apart_II21 from_II22 the_AT top_JJ row_NN1 of_IO &lsqb;_( formula_NN1 &rsqb;_) which_DDQ by_II inspection_NN1 is_VBZ identical_JJ with_IW that_DD1 of_IO A_ZZ1 ,_, we_PPIS2 have_VH0 here_RL 12_MC equations_NN2 ad_RA 12_MC unknowns_NN2 ._. 
From_II the_AT first_MD column_NN1 of_IO the_AT product_NN1 we_PPIS2 have_VH0 &lsqb;_( formula_NN1 &rsqb;_) which_DDQ determine_VV0 p_ZZ1 ,_, q_ZZ1 ,_, r_ZZ1 ._. 
The_AT second_MD column_NN1 yields_NN2 &lsqb;_( formula_NN1 &rsqb;_) which_DDQ ,_, with_IW p_ZZ1 ,_, ._. 
q_ZZ1 ,_, r_ZZ1 known_VVN ,_, determine_VV0 in_II succession_NN1 &lsqb;_( formula_NN1 &rsqb;_) 2_MC ,_, s_ZZ1 ,_, t_ZZ1 ._. 
The_AT remaining_JJ unknowns_NN2 &lsqb;_( formula_NN1 &rsqb;_) 3_MC ,_, x33_FO ,_, n_ZZ1 and_CC the_AT last_MD column_NN1 of_IO R1_FO are_VBR found_VVN progressively_RR in_II the_AT same_DA way_NN1 ._. 
Note_VV0 ,_, however_RR ,_, that_CST A11_FO ,_, &lsqb;_( formula_NN1 &rsqb;_) 2_MC ,_, x33_FO ,_, ..._... are_VBR divisors_NN2 (_( &lsqb;_( formula_NN1 &rsqb;_) vots_VVZ )_) in_II this_DD1 procedure_NN1 ,_, so_CS21 that_CS22 a_AT1 &lsqb;_( formula_NN1 &rsqb;_) voting_NN1 strategy_NN1 may_VM be_VBI needed_VVN ._. 
Note_VV0 also_RR that_DD1 &lsqb;_( formula_NN1 &rsqb;_) ,_, so_RR &lsqb;_( formula_NN1 &rsqb;_) ;_; hence_RR ,_, if_CS A_ZZ1 is_VBZ non-singular_JJ ,_, the_AT diagonal_JJ elements_NN2 of_IO R1_FO must_VM all_DB be_VBI nonzero_NN1 ._. 
Indeed_RR ,_, this_DD1 program_NN1 is_VBZ very_RG suitable_JJ for_IF the_AT numerical_JJ evaluation_NN1 of_IO determinants_NN2 ._. 
The_AT &lsqb;_( formula_NN1 &rsqb;_) algorithm_NN1 resolves_VVZ A_ZZ1 into_II the_AT product_NN1 &lsqb;_( formula_NN1 &rsqb;_) as_CSA above_RL ,_, and_CC then_RT multiplies_VVZ the_AT factors_NN2 in_II reverse_JJ order_NN1 to_TO obtain_VVI a_AT1 new_JJ matrix_NN1 B._NP1 Since_CS &lsqb;_( formula_NN1 &rsqb;_) and_CC then_RT &lsqb;_( formula_NN1 &rsqb;_) is_VBZ a_AT1 similar_JJ transform_NN1 of_IO A._NP1 B_ZZ1 is_VBZ ,_, in_RR21 general_RR22 ,_, a_AT1 fully_RR populated_VVN matrix_NN1 of_IO which_DDQ the_AT last_MD column_NN1 is_VBZ that_DD1 of_IO R1_FO (_( thus_RR A14_FO reappears_VVZ in_II B_ZZ1 )_) and_CC the_AT last_MD row_VV0 that_DD1 of_IO L1_FO multiplied_VVN by_II x44_FO ._. 
When_CS B_ZZ1 is_VBZ found_VVN ,_, it_PPH1 is_VBZ in_II turn_NN1 resolved_VVN into_II the_AT product_NN1 L2R2_FO and_CC &lsqb;_( formula_NN1 &rsqb;_) evaluated_VVD ,_, and_RR31 so_RR32 on_RR33 ._. 
As_II the_AT succession_NN1 of_IO similar_JJ transforms_VVZ proceeds_NN2 ,_, it_PPH1 is_VBZ found_VVN that_CST (_( a_ZZ1 )_) the_AT elements_NN2 below_II the_AT diagonal_JJ become_VV0 progressively_RR smaller_JJR ,_, (_( b_ZZ1 )_) the_AT diagonal_JJ elements_NN2 tend_VV0 to_II the_AT eigenvalues_NN2 ,_, in_II descending_JJ order_NN1 or_CC modulus_NN1 down_II the_AT diagonal_JJ ._. 
When_CS the_AT process_NN1 is_VBZ well_RR established_VVN ,_, it_PPH1 is_VBZ found_VVN that_CST (_( with_IW n_ZZ1 =_FO 4_MC as_CSA our_APPGE model_NN1 )_) the_AT element_NN1 &lsqb;_( formula_NN1 &rsqb;_) becomes_VVZ approximately_RR &lsqb;_( formula_NN1 &rsqb;_) in_II &lsqb;_( formula_NN1 &rsqb;_) ._. 
When_CS this_DD1 element_NN1 becomes_VVZ very_RG small_JJ &lsqb;_( formula_NN1 &rsqb;_) tends_VVZ to_TO decrease_VVI by_II the_AT factor_NN1 &lsqb;_( formula_NN1 &rsqb;_) and_CC &lsqb;_( formula_NN1 &rsqb;_) by_II &lsqb;_( formula_NN1 &rsqb;_) and_RR31 so_RR32 on_RR33 ._. 
Accordingly_RR ,_, if_CS the_AT eigenvalues_NN2 of_IO A_ZZ1 are_VBR widely_RR spaced_VVN ,_, convergence_NN1 of_IO the_AT &lsqb;_( formula_NN1 &rsqb;_) method_NN1 is_VBZ fairly_RR ra&lsqb;formula&rsqb;d_VV0 ;_; if_CS ,_, however_RR ,_, there_EX is_VBZ a_AT1 pair_NN or_CC group_NN1 of_IO nearly-equal_JJ eigenvalues_NN2 ,_, then_RT very_RG many_DA2 iterations_NN2 may_VM be_VBI required_VVN before_II the_AT end_NN1 result_NN1 is_VBZ achieved_VVN ._. 
This_DD1 is_VBZ an_AT1 upper_JJ triangular_JJ matrix_NN1 T_ZZ1 ,_, of_IO which_DDQ the_AT diagonal_JJ elements_NN2 are_VBR the(real)_JJ eigenvalues_NN2 in_II descending_JJ order_NN1 of_IO modulus_NN1 ,_, and_CC which_DDQ still_RR retains_VVZ the_AT element_NN1 A14_FO in_II the_AT top_JJ right_JJ position_NN1 ._. 
If_CS A_ZZ1 has_VHZ a_AT1 pair_NN of_IO conjugate_NN1 complex_JJ eigenvalues_NN2 ,_, then_RT T_ZZ1 is_VBZ not_XX strictly_RR upper_JJ triangular_JJ :_: the_AT diagonal_JJ includes_VVZ a_AT1 block_NN1 of_IO order_NN1 2_MC such_II21 as_II22 &lsqb;_( formula_NN1 &rsqb;_) with_IW &lsqb;_( formula_NN1 &rsqb;_) on_II the_AT diagonal_JJ of_IO T_ZZ1 and_CC &lsqb;_( formula_NN1 &rsqb;_) below_RL it_PPH1 ._. 
In_II this_DD1 case_NN1 ,_, the_AT solution_NN1 of_IO &lsqb;_( formula_NN1 &rsqb;_) yields_VVZ the_AT complex_JJ eigenvalues._NNU the_AT numbers_NN2 ,_, &lsqb;_( formula_NN1 &rsqb;_) not_XX in_RR21 general_RR22 constant_JJ under_II transformation_NN1 ;_; but_CCB if_CS the_AT iterations_NN2 have_VH0 proceeded_VVN so_RG far_RR that_CST I_ZZ1 is_VBZ in_II the_AT quasi-triangular_JJ form_NN1 ,_, the_AT trace_NN1 and_CC determinant_NN1 of_IO (_( 3_MC )_) are_VBR invariant_JJ ,_, so_CS21 that_CS22 the_AT complex_JJ roots_NN2 are_VBR fixed_VVN ._. 
As_II the_AT LR_NP1 transformations_NN2 proceed_VV0 ,_, the_AT most_RRT ra&lsqb;formula&rsqb;d_VV0 convergence_NN1 is_VBZ to_II the_AT smallest_JJT eigenvalue_NN1 in_II the_AT bottom_JJ right-hand_JJ corner_NN1 ._. 
Indeed_RR ,_, if_CS there_EX is_VBZ a_AT1 zero_NN1 eigenvalue_NN1 ,_, it_PPH1 appears_VVZ in_II the_AT first_MD transformation_NN1 ,_, i.e._REX in_II B._NP1 It_PPH1 follows_VVZ that_DD1 shifting_NN1 (_( see_VV0 2.7.4_MC )_) can_VM be_VBI used_VVN with_IW great_JJ advantage_NN1 to_TO accelerate_VVI convergence_NN1 ._. 
When_CS the_AT element_NN1 in_II the_AT bottom_JJ right-hand_JJ corner_NN1 has_VHZ reached_VVN a_AT1 settled_JJ value_NN1 (_( i.e._REX the_AT eigenvalue_NN1 )_) ,_, we_PPIS2 can_VM deflate_VVI the_AT transform_NN1 ,_, by_II omission_NN1 of_IO its_APPGE last_MD row_NN1 and_CC column_NN1 ,_, before_II continuing_VVG the_AT iterations_NN2 with_IW a_AT1 matrix_NN1 of_IO reduced_JJ order._NN1 ,_, /div4_VV0 ._. 
Example_NN1 1_MC1 We_PPIS2 choose_VV0 the_AT deflated_JJ matrix_NN1 i_ZZ1 (_( 2.7.2.2_MC )_) as_CSA A_ZZ1 :_: &lsqb;_( formula_NN1 &rsqb;_) Since_CS &lsqb;_( formula_NN1 &rsqb;_) ,_, the_AT bottom_JJ right-hand_JJ element_NN1 of_IO &lsqb;_( formula_NN1 &rsqb;_) vanishes_VVZ ,_, as_CSA it_PPH1 should_VM ._. 
We_PPIS2 now_RT evaluate_VV0 the_AT product_NN1 in_II reverse_JJ order_NN1 :_: &lsqb;_( formula_NN1 &rsqb;_) The_AT reader_NN1 is_VBZ invited_VVN to_TO check_VVI that_DD1 ,_, if_CS B_ZZ1 is_VBZ resolved_VVN as_CSA it_PPH1 stands_VVZ into_II &lsqb;_( formula_NN1 &rsqb;_) then_RT C_ZZ1 =_FO &lsqb;_( formula_NN1 &rsqb;_) will_VM also_RR have_VHI its_APPGE last_MD row_NN1 null_JJ ,_, the_AT zero_NN1 eigenvalue_NN1 thus_RR repeating_VVG ._. 
We_PPIS2 may_VM therefore_RR deflate_VVI B_ZZ1 ,_, omitting_VVG its_APPGE last_MD row_NN1 and_CC column_NN1 and_CC continue_VV0 the_AT iterations_NN2 with_IW the_AT leading_JJ first_MD minor_NN1 of_IO B._NP1 It_PPH1 is_VBZ found_VVN that_CST ,_, after_II 22_MC more_DAR iterations_NN2 ,_, working_VVG to_II six_MC decimal_JJ places_NN2 ,_, &lsqb;_( formula_NN1 &rsqb;_) Thus_RR ,_, the_AT eigenvalues_NN2 of_IO A_ZZ1 are_VBR ,_, in_II descending_JJ order_NN1 of_IO modulus_NN1 ,_, 12.5_MC ,_, 5.0_MC ,_, 2.5_MC ,_, 0_MC ._. 
The_AT whole_JJ operation_NN1 may_VM in_II fact_NN1 be_VBI summarised_VVN as_CSA LT_NNB =_FO AL_NP1 :_: &lsqb;_( formula_NN1 &rsqb;_) Recovery_NN1 of_IO the_AT eigenvectors_NN2 in_II the_AT LR_NP1 procedure_NN1 is_VBZ not_XX straightforward_JJ ,_, especially_RR if_CS deflation_NN1 has_VHZ been_VBN used_VVN ._. 
Since_CS the_AT eigenvalues_NN2 are_VBR known_VVN ,_, probably_RR the_AT simplest_JJT method_NN1 is_VBZ to_TO use_VVI (_( 2.6.4_MC )_) ._. 
However_RR ,_, if_CS we_PPIS2 have_VH0 not_XX deflated_VVN ,_, then_RT clearly_RR &lsqb;_( formula_NN1 &rsqb;_) leads_VVZ to_II (_( see_VV0 (_( 5_MC )_) above_RL )_) &lsqb;_( formula_NN1 &rsqb;_) We_PPIS2 now_RT require_VV0 to_TO transform_VVI T_ZZ1 to_II &lsqb;_( formula_NN1 &rsqb;_) ._. 
If_CS TP_NP1 =_FO PA_NN1 ,_, then_RT &lsqb;_( formula_NN1 &rsqb;_) where_RRQ PA_NN1 has_VHZ been_VBN multiplied_VVN out_RP ._. 
Identification_NN1 of_IO the_AT elements_NN2 on_II each_DD1 side_NN1 ,_, beginning_VVG with_IW the_AT superdiagonal_JJ ,_, leads_VVZ to_II &lsqb;_( formula_NN1 &rsqb;_) giving_VVG a_AT1 ,_, d_ZZ1 ,_, f_ZZ1 ._. 
The_AT remaining_JJ unknowns_NN2 ,_, b_ZZ1 ,_, e_ZZ1 ,_, c_ZZ1 are_VBR found_VVN progressively_RR ._. 
With_IW P_ZZ1 determined_VVN in_II this_DD1 way_NN1 ,_, we_PPIS2 have_VH0 &lsqb;_( formula_NN1 &rsqb;_) so_CS21 that_CS22 the_AT modal_JJ matrix_NN1 of_IO A_ZZ1 is_VBZ the_AT product_NN1 LP_NN1 ._. 
The_AT reader_NN1 is_VBZ invited_VVN to_TO evaluate_VVI LP_NN1 ,_, using_VVG the_AT numerical_JJ values_NN2 of_IO (_( 5_MC )_) in_II (_( 6_MC )_) ,_, and_CC in_RR21 particular_RR22 to_TO note_VVI that_CST the_AT eigenvector_NN1 belonging_VVG to_II the_AT zero_NN1 root_NN1 is_VBZ that_CST originally_RR associated_VVN with_IW the_AT eigenvalues_NN2 25_MC in_II (_( 2.7.2.6_MC )_) The_AT LR_NP1 procedure_NN1 is_VBZ thus_RR simple_JJ and_CC easily_RR programmed_VVN ._. 
However_RR ,_, in_II practice_NN1 ,_, it_PPH1 normally_RR requires_VVZ very_RG many_DA2 iterations_NN2 before_II convergence_NN1 is_VBZ achieved_VVN ;_; moreover_RR ,_, the_AT process_NN1 of_IO recovering_VVG eigenvectors_NN2 outlined_VVN above_RL is_VBZ apt_JJ to_TO be_VBI ill-conditioned_JJ ,_, and_CC some_DD other_JJ procedure_NN1 is_VBZ usually_RR to_TO be_VBI preferred._NNU 2.8.4_MC Pre-reduction_JJ to_II upper_JJ Hessenberg_NP1 form_NN1 As_CSA we_PPIS2 have_VH0 seen_VVN ,_, the_AT LR_NP1 procedure_NN1 (_( and_CC the_AT QR_NP1 method_NN1 ,_, to_TO be_VBI discussed_VVN shortly_RR )_) annihilates_VVZ the_AT elements_NN2 of_IO A_ZZ1 below_II the_AT diagonal_JJ ._. 
This_DD1 is_VBZ ,_, however_RR ,_, usually_RR a_AT1 very_RG long_JJ process_NN1 ;_; and_CC while_CS it_PPH1 can_VM be_VBI accelerated_VVN by_II shifting_VVG and_CC deflation_NN1 ,_, it_PPH1 is_VBZ in_II fact_NN1 better_RRR to_TO depopulate_VVI A_ZZ1 below_II the_AT diagonal_JJ ,_, as_RG far_RR as_CSA possible_JJ ,_, as_CSA an_AT1 initial_JJ step_NN1 ._. 
An_AT1 upper_JJ Hessenberg_NP1 matrix_NN1 has_VHZ only_JJ zeros_NN2 below_II its_APPGE infradiagonal_JJ ;_; and_CC it_PPH1 is_VBZ possible_JJ to_TO transform_VVI any_DD matrix_NN1 A_ZZ1 to_II this_DD1 form_NN1 by_II a_AT1 similar_JJ transformation_NN1 ._. 
This_DD1 is_VBZ the_AT best_RRT that_DD1 can_VM be_VBI done_VDN to_TO depopulate_VVI a_AT1 matrix_NN1 below_II its_APPGE diagonal_JJ ;_; if_CS one_PN1 could_VM remove_VVI infradiagonal_JJ elements_NN2 as_RR21 well_RR22 ,_, there_EX would_VM be_VBI no_AT problem_NN1 in_II finding_VVG eigenvalues_NN2 and_CC vectors_NN2 of_IO a_AT1 matrix_NN1 ._. 
We_PPIS2 illustrate_VV0 reduction_NN1 to_II upper_JJ Hessenberg_NP1 form_NN1 with_IW a_AT1 general_JJ matrix_NN1 of_IO order_NN1 4_MC ;_; it_PPH1 is_VBZ clear_JJ that_CST the_AT procedure_NN1 applies_VVZ to_II matrices_NN2 of_IO any_DD order_NN1 greater_JJR than_CSN 2_MC ._. 
We_PPIS2 write_VV0 AK_NP1 =_FO KH_NP1 as_CSA &lsqb;_( formula_NN1 &rsqb;_) Just_RR as_CSA we_PPIS2 have_VH0 done_VDN earlier_RRR ,_, we_PPIS2 can_VM evaluate_VVI the_AT n2_FO unknowns_NN2 progressively_RR ._. 
The_AT first_MD column_NN1 of_IO the_AT product_NN1 gives_VVZ &lsqb;_( formula_NN1 &rsqb;_) which_DDQ determine_VV0 h11_FO ,_, h21_FO ,_, p_ZZ1 ,_, q_ZZ1 (_( note_VV0 that_CST here_RL ,_, also_RR ,_, a_AT1 &lsqb;_( formula_NN1 &rsqb;_) voting_NN1 strategy_NN1 may_VM be_VBI needed_VVN )_) ._. 
From_II the_AT second_MD column_NN1 &lsqb;_( formula_NN1 &rsqb;_) which_DDQ yield_VV0 in_II succession_NN1 &lsqb;_( formula_NN1 &rsqb;_) and_RR31 so_RR32 on_RR33 ._. 
Thus_RR ,_, by_II progressive_JJ substitution_NN1 ,_, we_PPIS2 arrive_VV0 at_II the_AT Hessenberg_NP1 matrix_NN1 &lsqb;_( formula_NN1 &rsqb;_) ;_; it_PPH1 possesses_VVZ the_AT same_DA eigenvalues_NN2 as_CSA A._NP1 If_CS we_PPIS2 apply_VV0 the_AT LR_NP1 or_CC QR_NP1 algorithms_NN2 to_II H_ZZ1 ,_, the_AT Hessenberg_NP1 form_NN1 is_VBZ retained_VVN at_II each_DD1 step_NN1 ;_; thus_RR ,_, instead_II21 of_II22 removing_VVG all_DB the_AT elements_NN2 below_II the_AT diagonal_JJ ,_, we_PPIS2 have_VH0 to_TO remove_VVI only_RR those_DD2 in_II the_AT infradiagonal_JJ ._. 
Much_DA1 computer_NN1 arithmetic_NN1 and_CC time_NNT1 is_VBZ thus_RR saved_VVN ._. 
Transformation_NN1 of_IO A_ZZ1 to_II H_ZZ1 therefore_RR greatly_RR improves_VVZ the_AT LR_NP1 algorithm_NN1 ;_; we_PPIS2 have_VH0 introduced_VVN it_PPH1 here_RL ,_, however_RR ,_, because_CS for_IF the_AT QR_NP1 method_NN1 it_PPH1 is_VBZ a_AT1 sine_NN131 qua_NN132 non_NN133 ._. 
We_PPIS2 may_VM add_VVI here_RL ,_, in_II passing_NN1 ,_, that_CST it_PPH1 is_VBZ possible_JJ by_II simple_JJ substitutions_NN2 to_TO reduce_VVI any_DD matrix_NN1 A_ZZ1 to_II tridiagonal_JJ form_NN1 populated_VVN only_RR in_II the_AT superdiagonal_JJ ,_, diagonal_JJ and_CC infradiagonal_JJ ._. 
However_RR ,_, for_IF our_APPGE present_JJ purposes_NN2 this_DD1 is_VBZ unnecessary_JJ ;_; and_CC in_II any_DD event_NN1 it_PPH1 often_RR involves_VVZ ill-conditioned_JJ equations_NN2 and_CC so_RR lacks_VVZ accuracy_NN1 ._. 
Example_NN1 1_MC1 We_PPIS2 again_RT choose_VV0 the_AT matrix_NN1 A_ZZ1 of_IO (_( 2.8.3.4_MC )_) ._. 
The_AT reader_NN1 is_VBZ invited_VVN to_TO check_VVI that_DD1 ,_, if_CS A_ZZ1 is_VBZ transformed_VVN as_CSA in_II (_( 1_MC1 )_) ,_, then_RT &lsqb;_( formula_NN1 &rsqb;_) If_CS we_PPIS2 now_RT treat_VV0 H_ZZ1 as_CSA our_APPGE basic_JJ starting_NN1 matrix_NN1 A_ZZ1 and_CC resolve_VV0 it_PPH1 into_II L1R1_FO then_RT &lsqb;_( formula_NN1 &rsqb;_) As_CSA will_VM be_VBI seen_VVN from_II &lsqb;_( formula_NN1 &rsqb;_) ,_, resolution_NN1 of_IO an_AT1 upper_JJ Hessenberg_NP1 matrix_NN1 requires_VVZ an_AT1 upper_JJ Hessenberg_NP1 L1_FO ,_, i.e._REX a_AT1 matrix_NN1 with_IW zero_MC elements_NN2 everywhere_RL except_CS in_II the_AT diagonal_JJ and_CC infradiagonal_JJ ._. 
Both_DB2 the_AT resolution_NN1 of_IO A_ZZ1 into_II L1R1_FO and_CC the_AT subsequent_JJ evaluation_NN1 of_IO &lsqb;_( formula_NN1 &rsqb;_) (_( B_ZZ1 is_VBZ also_RR of_IO upper_JJ Hessenberg_NP1 form_NN1 )_) are_VBR thus_RR much_RR simpler_JJR than_CSN when_CS A_ZZ1 is_VBZ fully_RR populated._NNU 2.8.5_MC The_AT QR_NP1 algorithm_NN1 We_PPIS2 assume_VV0 that_CST the_AT matrix_NN1 A_ZZ1 is_VBZ already_RR in_II upper_JJ Hessenberg_NP1 form_NN1 ._. 
Then_RT ,_, in_II parallel_NN1 with_IW the_AT LR_NP1 algorithm_NN1 ,_, we_PPIS2 write_VV0 &lsqb;_( formula_NN1 &rsqb;_) where_RRQ &lsqb;_( formula_NN1 &rsqb;_) ,_, as_CSA before_RT ,_, is_VBZ upper_JJ triangular_JJ ._. 
However_RR ,_, here_RL we_PPIS2 do_VD0 not_XX resolve_VVI A_ZZ1 into_II &lsqb;_( formula_NN1 &rsqb;_) instead_RR ,_, we_PPIS2 require_VV0 that_CST &lsqb;_( formula_NN1 &rsqb;_) shall_VM be_VBI a_AT1 orthogonal_JJ matrix_NN1 :_: &lsqb;_( formula_NN1 &rsqb;_) Thus_RR &lsqb;_( formula_NN1 &rsqb;_) is_VBZ our_APPGE similar_JJ transformation_NN1 of_IO A_ZZ1 into_II B._NP1 We_PPIS2 then_RT transform_VV0 B_ZZ1 into_II C_NP1 :_: &lsqb;_( formula_NN1 &rsqb;_) and_RR31 so_RR32 on_RR33 ,_, until_CS the_AT transform_NN1 is_VBZ eventually_RR an_AT1 upper_JJ triangular_JJ matrix_NN1 having_VHG the_AT eigenvalues_NN2 of_IO A_ZZ1 in_II its_APPGE diagonal_JJ ._. 
It_PPH1 remains_VVZ to_TO explain_VVI how_RRQ Qi_NP1 is_VBZ chosen_VVN ._. 
There_EX are_VBR various_JJ possibilities_NN2 ;_; the_AT most_RGT popular_JJ methods_NN2 are_VBR those_DD2 of_IO Givens_NP1 and_CC Householder_NP1 (_( 6_MC )_) ._. 
Here_RL we_PPIS2 describe_VV0 Givens_58 '_GE method_NN1 ;_; and_CC as_CSA before_RT ,_, for_IF simplicity_NN1 of_IO exposition_NN1 ,_, we_PPIS2 choose_VV0 n_ZZ1 =_FO 4_MC ._. 
Let_VV0 the_AT matrix_NN1 ,_, pre-reduced_NN1 to_II upper_JJ Hessenberg_NP1 form_NN1 ,_, be_VBI &lsqb;_( formula_NN1 &rsqb;_) Also_RR ,_, let_VV0 &lsqb;_( formula_NN1 &rsqb;_) where_RRQ &lsqb;_( formula_NN1 &rsqb;_) ._. 
Then_RT if_CS we_PPIS2 evaluate_VV0 &lsqb;_( formula_NN1 &rsqb;_) as_II a_AT1 chain_NN1 ,_, beginning_VVG by_II premultiplying_VVG A_ZZ1 by_II the_AT last_MD of_IO the_AT three_MC matrices_NN2 in_II (_( 4_MC )_) ,_, and_CC if_CS also_RR we_PPIS2 take_VV0 &lsqb;_( formula_NN1 &rsqb;_) then_RT A_ZZ1 is_VBZ first_MD replaced_VVN by_II a_AT1 matrix_NN1 in_II which_DDQ the_AT leading_JJ element_NN1 of_IO the_AT infradiagonal_JJ vanishes_VVZ ._. 
The_AT top_NN1 two_MC rows_NN2 of_IO A_ZZ1 are_VBR altered_VVN ;_; in_RR21 particular_RR22 ,_, &lsqb;_( formula_NN1 &rsqb;_) 2_MC is_VBZ replaced_VVN by_II &lsqb;_( formula_NN1 &rsqb;_) ._. 
In_II the_AT next_MD multiplication_NN1 we_PPIS2 take_VV0 &lsqb;_( formula_NN1 &rsqb;_) and_CC then_RT the_AT second_MD element_NN1 in_II the_AT infradiagonal_JJ vanishes_VVZ ._. 
Finally_RR ,_, with_IW A33_FO now_RT altered_VVN to_II &lsqb;_( formula_NN1 &rsqb;_) we_PPIS2 put_VV0 &lsqb;_( formula_NN1 &rsqb;_) and_CC complete_VV0 the_AT chain_NN1 ._. 
The_AT result_NN1 is_VBZ the_AT upper_JJ triangular_JJ matrix_NN1 R1_FO ._. 
We_PPIS2 now_RT have_VH0 to_TO complete_VVI the_AT similar_JJ transformation_NN1 by_II postmultiplying_VVG by_II Q1_FO ;_; the_AT result_NN1 ,_, &lsqb;_( formula_NN1 &rsqb;_) is_VBZ again_RT of_IO upper_JJ Hessenberg_NP1 form_NN1 ._. 
We_PPIS2 now_RT repeat_VV0 the_AT cycle_NN1 to_TO obtain_VVI &lsqb;_( formula_NN1 &rsqb;_) and_RR31 so_RR32 on_RR33 ._. 
As_CSA before_RT ,_, it_PPH1 is_VBZ found_VVN that_CST the_AT infradiagonal_JJ elements_NN2 become_VV0 smaller_JJR at_II each_DD1 step_NN1 until_CS the_AT similar_JJ transform_NN1 of_IO A_ZZ1 is_VBZ ultimately_RR upper_JJ triangular_JJ (_( or_CC quasi-triangular_JJ if_CS A_ZZ1 has_VHZ complex_JJ eigenvalues_NN2 )_) ;_; i.e._REX a_AT1 matrix_NN1 T_ZZ1 given_VVN by_II &lsqb;_( formula_NN1 &rsqb;_) The_AT following_JJ point_NN1 is_VBZ important_JJ ._. 
In_II our_APPGE treatment_NN1 above_RL ,_, B_ZZ1 is_VBZ the_AT product_NN1 of_IO a_AT1 chain_NN1 of_IO seven_MC matrices_NN2 ._. 
When_CS the_AT multiplications_NN2 are_VBR carried_VVN out_RP in_II the_AT order_NN1 indicated_VVD ,_, the_AT upper_JJ Hessenberg_NP1 form_NN1 is_VBZ maintained_VVN throughout_RL ,_, If_CS ,_, however_RR ,_, we_PPIS2 begin_VV0 by_II multiplying_VVG the_AT central_JJ three_MC matrices_NN2 ,_, the_AT upper_JJ Hessenberg_NP1 form_NN1 is_VBZ lost_VVN ,_, with_IW a_AT1 accompanying_JJ loss_NN1 of_IO simplicity_NN1 ._. 
It_PPH1 is_VBZ now_RT apparent_JJ why_RRQ A_AT1 must_NN1 be_VBI pre-reduced_JJ to_II upper_JJ Hessenberg_NP1 form_NN1 ._. 
Equation_NN1 (_( 4_MC )_) employs_VVZ one_MC1 matrix_NN1 for_IF each_DD1 nonzero_NN1 element_NN1 below_II the_AT diagonal_JJ three_MC in_II this_DD1 case_NN1 ;_; n_ZZ1 1_MC1 in_RR21 general_RR22 ._. 
There_EX are_VBR thus_RR 2n_FO 2_MC multiplications_NN2 in_II each_DD1 iteration_NN1 ._. 
But_CCB if_CS A_ZZ1 were_VBDR fully_RR populated_VVN ,_, the_AT number_NN1 of_IO multiplications_NN2 would_VM be_VBI n_ZZ1 (_( n_ZZ1 1_MC1 )_) ._. 
Example_NN1 1_MC1 The_AT reader_NN1 is_VBZ invited_VVN to_TO work_VVI through_II this_DD1 example_NN1 ;_; we_PPIS2 avoid_VV0 repetitive_JJ labour_NN1 by_II the_AT use_NN1 of_IO hindsight_NN1 ._. 
Consider_VV0 the_AT matrix_NN1 defined_VVN in_II (_( 2.7.6.12_MC )_) and_CC reduce_VV0 it_PPH1 ,_, as_CSA in_II (_( 2.8.4.1_MC )_) ,_, to_II upper_JJ Hessenberg_NP1 form_NN1 ._. 
The_AT result_NN1 is_VBZ a_AT1 new_JJ matrix_NN1 &lsqb;_( formula_NN1 &rsqb;_) We_PPIS2 now_RT ,_, with_IW hindsight_NN1 ,_, use_VV0 a_AT1 shift_NN1 of_IO 0.2_MC ;_; i.e._REX we_PPIS2 work_VV0 with_IW the_AT matrix_NN1 A_ZZ1 0.2I_FO ._. 
Shifts_NN2 are_VBR always_RR used_VVN to_TO improve_VVI convergence_NN1 :_: this_DD1 particular_JJ shift_NN1 requires_VVZ only_RR one_MC1 step_NN1 ._. 
The_AT matrix_NN1 is_VBZ now_RT &lsqb;_( formula_NN1 &rsqb;_) and_CC after_CS the_AT six_MC multiplications_NN2 implied_VVN (_( see_VV0 (_( 4_MC )_) )_) in_II &lsqb;_( formula_NN1 &rsqb;_) it_PPH1 becomes_VVZ &lsqb;_( formula_NN1 &rsqb;_) Thus_RR A_ZZ1 0.2I_FO has_VHZ a_AT1 zero_NN1 eigenvalue_NN1 ;_; we_PPIS2 recover_VV0 B_ZZ1 by_II adding_VVG 0.2I_FO and_CC can_VM then_RT deflate_VVI and_CC consider_VVI the_AT reduced_JJ matrix_NN1 &lsqb;_( formula_NN1 &rsqb;_) Once_RR21 again_RR22 ,_, to_TO avoid_VVI repetitive_JJ labour_NN1 in_II the_AT example_NN1 ,_, we_PPIS2 use_VV0 hindsight_NN1 :_: we_PPIS2 now_RT apply_VV0 a_AT1 shift_NN1 of_IO 0.4_MC and_CC consider_VV0 &lsqb;_( formula_NN1 &rsqb;_) We_PPIS2 require_VV0 only_RR one_MC1 step_NN1 (_( four_MC multiplications_NN2 )_) to_TO reduce_VVI this_DD1 to_II &lsqb;_( formula_NN1 &rsqb;_) We_PPIS2 now_RT recover_VV0 &lsqb;_( formula_NN1 &rsqb;_) by_II adding_VVG 0.4I_FO and_CC deflate_VV0 by_II omission_NN1 of_IO the_AT last_MD row_NN1 and_CC column_NN1 :_: &lsqb;_( formula_NN1 &rsqb;_) In_II any_DD similar_JJ transform_NN1 of_IO &lsqb;_( formula_NN1 &rsqb;_) the_AT trace_NN1 and_CC determinant_NN1 are_VBR unaltered_JJ ._. 
The_AT two_MC eigenvalues_NN2 of_IO &lsqb;_( formula_NN1 &rsqb;_) are_VBR thus_RR determined_VVN by_II &lsqb;_( formula_NN1 &rsqb;_) The_AT required_JJ eigenvalues_NN2 are_VBR thus_RR 0.2_MC (_( bottom_JJ right-hand_JJ element_NN1 of_IO B_ZZ1 )_) ,_, 0.4_MC (_( bottom_JJ right-hand_JJ element_NN1 of_IO &lsqb;_( formula_NN1 &rsqb;_) )_) and_CC &lsqb;_( formula_NN1 &rsqb;_) ._. 
The_AT convergence_NN1 properties_NN2 of_IO the_AT QR_NP1 algorithm_NN1 are_VBR superior_JJ to_II those_DD2 of_IO LR_NP1 ,_, particularly_RR when_CS shifts_NN2 are_VBR used_VVN and_CC deflation_NN1 is_VBZ performed_VVN whenever_RRQV possible_JJ ._. 
If_CS complex_JJ eigenvalues_NN2 exist_VV0 ,_, &quot;_" double_JJ shifts_NN2 '_GE (_( see_VV0 Wilkinson_NP1 (_( 6_MC )_) )_) may_VM be_VBI made_VVN and_CC two_MC QR_NP1 steps_NN2 performed_VVD simultaneously_RR to_TO give_VVI rapid_JJ convergence_NN1 while_CS all_DB arithmetic_NN1 is_VBZ kept_VVN real_JJ ._. 
The_AT QR_NP1 method_NN1 is_VBZ thus_RR very_RG powerful_JJ and_CC of_IO quite_RG general_JJ applicability_NN1 ;_; it_PPH1 is_VBZ probably_RR the_AT most_RGT popular_JJ technique_NN1 on_II modern_JJ main-frame_NN1 computers_NN2 for_IF the_AT evaluation_NN1 of_IO eigenvalues_NN2 of_IO non-symmetric_JJ matrices_NN2 ._. 
On_II the_AT other_JJ hand_NN1 ,_, recovery_NN1 of_IO eigenvectors_NN2 ,_, though_CS possible_JJ ,_, is_VBZ very_RG difficult_JJ and_CC often_RR inaccurate_JJ ,_, and_CC some_DD quite_RG different_JJ routine_NN1 (_( e.g._REX (_( 2.6.4_MC )_) ,_, when_CS the_AT eigenvalues_NN2 have_VH0 been_VBN found_VVN by_II QR_NP1 )_) is_VBZ recommended._NNU ,_, 2.9_MC MATRIX_NN1 PENCILS_NN2 Our_APPGE discussion_NN1 of_IO this_DD1 to&lsqb;formula&rsqb;c_NN1 is_VBZ very_RG limited_JJ ,_, though_CS we_PPIS2 look_VV0 again_RT at_II the_AT subject_NN1 ,_, also_RR briefly_RR ,_, in_II 2.10._MC 2.9.1_MC Eigenvalues_NN2 and_CC vectors_NN2 of_IO matrix_NN1 pencils_NN2 Any_DD lambda-matrix_NN1 of_IO the_AT form_NN1 &lsqb;_( formula_NN1 &rsqb;_) C_ZZ1 B_ZZ1 ,_, which_DDQ can_VM in_RR21 general_RR22 be_VBI rectangular_JJ ,_, *is_FO described_VVN as_II a_AT1 matrix_NN1 pencil_NN1 (_( see_VV0 1.15_MC )_) ._. 
If_CS C_ZZ1 is_VBZ square_JJ ,_, of_IO order_NN1 n_ZZ1 ,_, and_CC non-singular_NN1 ,_, the_AT pencil_NN1 is_VBZ described_VVN as_CSA regular_JJ ,_, since_CS in_II the_AT polynomial_NN1 equation_NN1 obtained_VVN by_II expansion_NN1 of_IO &lsqb;_( formula_NN1 &rsqb;_) the_AT coefficient_NN1 &lsqb;_( formula_NN1 &rsqb;_) which_DDQ does_VDZ not_XX vanish_VVI ,_, so_CS21 that_CS22 there_EX are_VBR n_ZZ1 eigenvalues_NN2 &lsqb;_( formula_NN1 &rsqb;_) i_ZZ1 which_DDQ satisfy_VV0 (_( 1_MC1 )_) ._. 
It_PPH1 is_VBZ clear_JJ that_CST these_DD2 eigenvalues_NN2 belong_VV0 also_RR to_II the_AT matrix_NN1 &lsqb;_( formula_NN1 &rsqb;_) ._. 
For_IF each_DD1 root_NN1 &lsqb;_( formula_NN1 &rsqb;_) i_ZZ1 there_EX will_VM be_VBI at_RR21 least_RR22 one_MC1 linear_JJ relation_NN1 between_II the_AT columns_NN2 of_IO the_AT pencil_NN1 ,_, so_CS21 that_CS22 we_PPIS2 may_VM write_VVI &lsqb;_( formula_NN1 &rsqb;_) and_CC &lsqb;_( formula_NN1 &rsqb;_) is_VBZ an_AT1 eigenvector_NN1 of_IO the_AT pencil_NN1 ._. 
The_AT pencil_NN1 is_VBZ described_VVN as_CSA simple_JJ if_CS there_EX are_VBR n_ZZ1 independent_JJ vectors_NN2 &lsqb;_( formula_NN1 &rsqb;_) so_CS21 that_CS22 we_PPIS2 may_VM write_VVI the_AT set_NN1 compendiously_RR as_CSA &lsqb;_( formula_NN1 &rsqb;_) where_RRQ &lsqb;_( formula_NN1 &rsqb;_) is_VBZ of_IO simple_JJ diagonal_JJ form_NN1 ,_, even_RR through_II it_PPH1 may_VM include_VVI multiple_JJ roots_NN2 ._. 
If_CS ,_, on_II the_AT other_JJ hand_NN1 ,_, one_MC1 or_CC more_DAR auxiliary_JJ vectors_NN2 are_VBR required_VVN ,_, so_CS21 that_CS22 the_AT spectral_JJ matrix_NN1 takes_VVZ the_AT form_NN1 &lsqb;_( formula_NN1 &rsqb;_) ,_, then_RT the_AT pencil_NN1 is_VBZ defective_JJ ._. 
In_RR21 particular_RR22 ,_, a_AT1 pencil_NN1 is_VBZ simple_JJ if_CS B_ZZ1 ,_, C_ZZ1 are_VBR real_JJ and_CC symmetric_JJ and_CC if_CS C_ZZ1 is_VBZ pos._NNU def_NN1 ._. 
This_DD1 proposition_NN1 is_VBZ proved_VVN in_II Corollary_NN1 1_MC1 of_IO Theorem_NN1 X_ZZ1 of_IO 1.22_MC ._. 
However_RR ,_, it_PPH1 does_VDZ not_XX follow_VVI that_CST if_CS B_ZZ1 ,_, C_ZZ1 are_VBR real_JJ and_CC symmetric_JJ ,_, the_AT pencil_NN1 is_VBZ simple_JJ ,_, or_CC its_APPGE roots_NN2 real_JJ ._. 
The_AT reader_NN1 is_VBZ invited_VVN to_TO study_VVI the_AT two_MC pencils_NN2 &lsqb;_( formula_NN1 &rsqb;_) Pencil_NN1 (_( i_ZZ1 )_) is_VBZ real_JJ and_CC symmetric_JJ ;_; it_PPH1 has_VHZ two_MC equal_JJ eigenvalues_NN2 &lsqb;_( formula_NN1 &rsqb;_) =_FO 1_MC1 and_CC the_AT spectral_JJ matrix_NN1 is_VBZ of_IO elementary_JJ Jordan_NP1 block_NN1 form_NN1 ;_; it_PPH1 is_VBZ defective_JJ ,_, with_IW only_RR the_AT single_JJ eigenvector_NN1 2,1_MC ._. 
Pencil_NN1 (_( ii_MC )_) also_RR has_VHZ real_JJ ,_, symmetric_JJ coefficients_NN2 ;_; but_CCB its_APPGE eigenvalues_NN2 are_VBR &lsqb;_( formula_NN1 &rsqb;_) ,_, with_IW independent_JJ vectors_NN2 1,1_MC +_FO 1_MC1 and_CC 1,1_MC -i_MC1 ._. 
It_PPH1 is_VBZ thus_RR simple_JJ ._. 
In_II both_RR (_( i_ZZ1 )_) and_CC (_( ii_MC )_) ,_, &lsqb;_( formula_NN1 &rsqb;_) is_VBZ negative_JJ ,_, and_CC so_CS C_ZZ1 is_VBZ not_XX pos._NNU def._NNU (_( see_VV0 1.22_MC ,_, Theorem_NN1 X_ZZ1 )_) ._. 
If_CS B_ZZ1 ,_, C_ZZ1 are_VBR real_JJ and_CC symmetric_JJ ,_, and_CC if_CS in_RR21 addition_RR22 C_ZZ1 is_VBZ pos._NNU def._NNU then_RT as_CSA in_II Theorem_NN1 VIII_MC of_IO 1.22_MC ,_, the_AT eigenvalues_NN2 are_VBR real_JJ and_CC finite_JJ and_CC the_AT eigenvectors_NN2 real_JJ ._. 
We_PPIS2 ow_VV0 restrict_VV0 our_APPGE attention_NN1 to_II simple_JJ pencils_NN2 ,_, where_CS symmetric_JJ or_CC not_XX ._. 
In_II parallel_NN1 with_IW (_( 2_MC )_) we_PPIS2 have_VH0 the_AT equation_NN1 &lsqb;_( formula_NN1 &rsqb;_) and_CC this_DD1 leads_VVZ to_II the_AT counterpart_NN1 of(3)_FO &lsqb;_( formula_NN1 &rsqb;_) Postmultiply_RR this_DD1 by_II X_ZZ1 and_CC use_NN1 (_( 3_MC )_) ;_; then_RT &lsqb;_( formula_NN1 &rsqb;_) For_IF simplicity_NN1 ,_, let_VV0 us_PPIO2 now_RT suppose_VVI all_DB roots_NN2 &lsqb;_( formula_NN1 &rsqb;_) i_ZZ1 to_TO be_VBI distinct_JJ ._. 
Then_RT &lsqb;_( formula_NN1 &rsqb;_) permutes_VVZ with_IW a_AT1 diagonal_JJ matrix_NN1 and_CC is_VBZ therefore_RR itself_PPX1 diagonal_JJ :_: &lsqb;_( formula_NN1 &rsqb;_) But_CCB (_( see_VV0 (_( 3_MC )_) )_) is_VBZ arbitrary_JJ to_II a_AT1 postmultiplying_JJ diagonal_JJ matrix_NN1 ;_; we_PPIS2 may_VM thus_RR write_VVI &lsqb;_( formula_NN1 &rsqb;_) and_CC then_RT from_II (_( 5_MC )_) &lsqb;_( formula_NN1 &rsqb;_) In_II the_AT special_JJ case_NN1 where_CS B_ZZ1 ,_, C_ZZ1 are_VBR symmetric_JJ ,_, we_PPIS2 may_VM evidently_RR identify_VVI Y_ZZ1 with_IW X._NP1 So_RG much_DA1 for_IF the_AT elementary_JJ properties_NN2 of_IO pencils_NN2 ._. 
We_PPIS2 turn_VV0 ow_NN1 to_II their_APPGE numerical_JJ treatment_NN1 ._. 
First_MD ,_, if_CS we_PPIS2 have_VH0 a_AT1 simple_JJ pencil_NN1 &lsqb;_( formula_NN1 &rsqb;_) C_ZZ1 B_ZZ1 ,_, we_PPIS2 may_VM premultiply_RR it_PPH1 by_II C-1_FO to_TO obtain_VVI &lsqb;_( formula_NN1 &rsqb;_) I_ZZ1 A_ZZ1 ,_, A_ZZ1 =_FO C-1B_FO ;_; we_PPIS2 may_VM then_RT obtain_VVI the_AT eigenvalues_NN2 and_CC vectors_NN2 of_IO the_AT pencil_NN1 by_II applying_VVG any_DD of_IO the_AT methods_NN2 of_IO 2.62.8_MC to_II A_ZZ1 ,_, and_CC often_RR this_DD1 is_VBZ the_AT simplest_JJT procedure_NN1 ._. 
However_RR ,_, cases_NN2 can_VM arise_VVI where_RRQ this_DD1 is_VBZ not_XX the_AT best_JJT approach_NN1 ._. 
For_REX21 example_REX22 ,_, suppose_VV0 C_ZZ1 to_TO be_VBI tridiagonal_JJ and_CC B_ZZ1 diagonal_JJ ._. 
Then_RT C-1_FO is_VBZ in_RR21 general_RR22 fully_RR populated_VVN ,_, and_CC so_RR would_VM be_VBI C-1B_FO ;_; the_AT simplicity_NN1 of_IO the_AT pencil_NN1 is_VBZ thus_RR lost_VVN in_II A._NP1 The_AT same_DA is_VBZ true_JJ when_CS C_ZZ1 ,_, B_ZZ1 are_VBR sparse_JJ ._. 
We_PPIS2 now_RT give_VV0 an_AT1 example_NN1 in_II which_DDQ C_NP1 ,_, B_ZZ1 are_VBR sparse_JJ and_CC symmetric_JJ ._. 
The_AT reader_NN1 is_VBZ invited_VVN to_TO study_VVI this_DD1 closely_RR ,_, since_CS it_PPH1 illustrates_VVZ more_DAR than_CSN the_AT possible_JJ treatment_NN1 of_IO a_AT1 pencil_NN1 ._. 
Example_NN1 1_MC1 Consider_VV0 the_AT pencil_NN1 &lsqb;_( formula_NN1 &rsqb;_) here_RL C_ZZ1 is_VBZ tridiagonal_JJ and_CC symmetric_JJ ;_; B_ZZ1 is_VBZ diagonal_JJ ._. 
Also_RR (_( though_CS this_DD1 is_VBZ immaterial_JJ for_IF our_APPGE present_JJ purpose_NN1 )_) both_DB2 are_VBR centrosymmetric_JJ ._. 
The_AT pencil_NN1 thus_RR has_VHZ particular_JJ simplicity_NN1 as_CSA it_PPH1 stands_VVZ ._. 
We_PPIS2 re_II required_JJ to_TO find_VVI the_AT eigenvalues_NN2 &lsqb;_( formula_NN1 &rsqb;_) i_ZZ1 which_DDQ satisfy_VV0 &lsqb;_( formula_NN1 &rsqb;_) ,_, and_CC the_AT corresponding_JJ vectors_NN2 ._. 
First_MD ,_, if_CS we_PPIS2 attempt_VV0 this_DD1 evaluating_VVG A_ZZ1 =_FO C-1B_FO we_PPIS2 find_VV0 &lsqb;_( formula_NN1 &rsqb;_) Thus_RR A_ZZ1 is_VBZ fully_RR populated_VVN and_CC is_VBZ not_XX symmetric_JJ (_( though_CS ,_, as_CSA it_PPH1 must_VM ,_, it_PPH1 remains_VVZ centrosymmetric_JJ )_) ._. 
We_PPIS2 therefore_RR return_VV0 to_II the_AT original_JJ form_NN1 (_( 8_MC )_) ._. 
A_AT1 possible_JJ way_NN1 of_IO dealing_VVG with_IW this_DD1 is_VBZ to_TO use_VVI the_AT location_NN1 method_NN1 (_( see_VV0 2.6.1_MC )_) ._. 
If_CS we_PPIS2 write_VV0 &lsqb;_( formula_NN1 &rsqb;_) then_RT &lsqb;_( formula_NN1 &rsqb;_) (_( since_CS P_ZZ1 is_VBZ of_IO order_NN1 4_MC )_) ;_; we_PPIS2 therefore_RR require_VV0 to_TO assign_VVI three_MC arbitrary_JJ values_NN2 to_II &lsqb;_( formula_NN1 &rsqb;_) and_CC then_RT ,_, by_II simple_JJ triangulation_NN1 ,_, evaluate_VV0 &lsqb;_( formula_NN1 &rsqb;_) for_IF each_DD1 of_IO these_DD2 ._. 
Since_CS the_AT product_NN1 of_IO the_AT roots_NN2 is_VBZ &lsqb;_( formula_NN1 &rsqb;_) ,_, we_PPIS2 may_VM choose_VVI &lsqb;_( formula_NN1 &rsqb;_) =_FO 2_MC ,_, 1_MC1 ,_, 1_MC1 as_CSA reasonable_JJ arbitrary_JJ values_NN2 ._. 
Then_RT we_PPIS2 find_VV0 ,_, using_VVG a_AT1 typical_JJ direct_JJ triangulation_NN1 &lsqb;_( formula_NN1 &rsqb;_) while_CS &lsqb;_( formula_NN1 &rsqb;_) ._. 
Fortuitously_RR ,_, we_PPIS2 have_VH0 found_VVN one_MC1 of_IO the_AT zeros_MC2 of&lsqb;formula&rsqb;_NN1 ._. 
Disregarding_VVG this_DD1 for_IF the_AT moment_NN1 ,_, we_PPIS2 write_VV0 &lsqb;_( formula_NN1 &rsqb;_) and_CC use_VV0 the_AT above_JJ values_NN2 ;_; we_PPIS2 find_VV0 &lsqb;_( formula_NN1 &rsqb;_) the_AT solution_NN1 of_IO which_DDQ is_VBZ &lsqb;_( formula_NN1 &rsqb;_) ._. 
Hence_RR &lsqb;_( formula_NN1 &rsqb;_) or_CC &lsqb;_( formula_NN1 &rsqb;_) To_TO find_VVI the_AT zeros_NN2 of_IO &lsqb;_( formula_NN1 &rsqb;_) (_( &lsqb;_( formula_NN1 &rsqb;_) )_) we_PPIS2 may_VM ,_, for_REX21 example_REX22 ,_, use_VV0 the_AT companion_NN1 matrix_NN1 (_( see_VV0 1.16_MC )_) ,_, which_DDQ ,_, in_II this_DD1 example_NN1 ,_, is_VBZ &lsqb;_( formula_NN1 &rsqb;_) the_AT eigenvalues_NN2 of_IO which_DDQ are_VBR the_AT zeros_NN2 of_IO (_( 10_MC )_) ._. 
We_PPIS2 may_VM find_VVI these_DD2 in_II the_AT usual_JJ way_NN1 by_II iteration_NN1 :_: we_PPIS2 repeatedly_RR premultiply_RR an_AT1 arbitrary_JJ column_NN1 ,_, say_VV0 &lsqb;_( formula_NN1 &rsqb;_) ,_, by_II M._NNU Successive_JJ columns_NN2 are_VBR then_RT given_VVN by_II Table_NN1 1_MC1 ._. 
We_PPIS2 observe_VV0 that_CST ,_, owing_II21 to_II22 the_AT particular_JJ form_NN1 of_IO M_ZZ1 ,_, we_PPIS2 need_VM calculate_VVI only_RR the_AT top_JJ element_NN1 of_IO each_DD1 column_NN1 ,_, in_II this_DD1 table_NN1 ;_; the_AT remaining_JJ elements_NN2 are_VBR those_DD2 of_IO the_AT previous_JJ column_NN1 ,_, one_MC1 step_VV0 down_RP ._. 
In_II fact_NN1 ,_, we_PPIS2 re_II only_RR solving_VVG the_AT regression_NN1 formula_NN1 &lsqb;_( formula_NN1 &rsqb;_) which_DDQ is_VBZ equivalent_JJ to_II the_AT iteration_NN1 with_IW M_ZZ1 ;_; this_DD1 is_VBZ in_II effect_NN1 Bernoulli_NP1 's_GE method_NN1 of_IO solution_NN1 of_IO &lsqb;_( formula_NN1 &rsqb;_) (_( &lsqb;_( formula_NN1 &rsqb;_) )_) =_FO 0_MC ._. 
To_II six_MC places_NN2 of_IO decimals_NN2 ,_, we_PPIS2 have_VH0 now_RT found_VVN &lsqb;_( formula_NN1 &rsqb;_) 1_MC1 6.464102_MC ._. 
Hence_RR &lsqb;_( formula_NN1 &rsqb;_) on_II extraction_NN1 of_IO the_AT first_MD factor_NN1 ._. 
If_CS we_PPIS2 set_VV0 up_RP a_AT1 regression_NN1 formula_NN1 (_( or_CC companion_NN1 matrix_NN1 )_) for_IF the_AT cubic_JJ ,_, we_PPIS2 find_VV0 the_AT next_MD factor_NN1 to_TO be_VBI (_( &lsqb;_( formula_NN1 &rsqb;_) +_FO 1_MC1 )_) ,_, as_CSA we_PPIS2 have_VH0 already_RR noted_VVN ._. 
In_II face_NN1 &lsqb;_( formula_NN1 &rsqb;_) so_CS21 that_CS22 the_AT eigenvalues_NN2 are_VBR &lsqb;_( formula_NN1 &rsqb;_) ._. 
We_PPIS2 now_RT look_VV0 briefly_RR at_II the_AT eigenvectors_NN2 ._. 
Typically_RR &lsqb;_( formula_NN1 &rsqb;_) which_DDQ is_VBZ of_RR21 course_RR22 singular_JJ ._. 
We_PPIS2 treat_VV0 it_PPH1 by_II the_AT method_NN1 of_IO Equation_NN1 (_( 2.6.4_MC )_) ,_, using_VVG the_AT first_MD three_MC rows_NN2 ,_, to_TO obtain_VVI with_IW x4_FO =1_FO ,_, &lsqb;_( formula_NN1 &rsqb;_) so_CS21 that_CS22 we_PPIS2 find_VV0 at_RR21 once_RR22 &lsqb;_( formula_NN1 &rsqb;_) ._. 
In_II this_DD1 way_NN1 ,_, with_IW the_AT eigenvalues_NN2 in_II order_NN1 of_IO modulus_NN1 ,_, we_PPIS2 find_VV0 &lsqb;_( formula_NN1 &rsqb;_) A_ZZ1 point_NN1 of_IO interest_NN1 may_VM be_VBI noted_VVN :_: the_AT individual_JJ vectors_NN2 are_VBR either_RR centrosymmetric_JJ or_CC centroskew_NN1 ,_, just_RR as_II a_AT1 symmetric_JJ structure_NN1 has_VHZ either_RR symmetric_JJ or_CC antisymmetric_JJ modes_NN2 of_IO vibration_NN1 ._. 
We_PPIS2 leave_VV0 it_PPH1 to_II the_AT reader_NN1 to_TO check_VVI that_CST (_( see_VV0 (_( 3_MC )_) )_) &lsqb;_( formula_NN1 &rsqb;_) and_CC also_RR that_DD1 &lsqb;_( formula_NN1 &rsqb;_) are_VBR diagonal_JJ ._. 
To_TO complete_VVI this_DD1 discussion_NN1 ,_, we_PPIS2 may_VM observe_VVI that_CST the_AT companion_NN1 matrix_NN1 ,_, or_CC its_APPGE equivalent_JJ the_AT regression_NN1 formula_NN1 ,_, may_VM be_VBI used_VVN to_TO find_VVI complex_JJ roots_NN2 of_IO a_AT1 determinantal_JJ equation_NN1 (_( see_VV0 1.7.6_MC )_) ._. 
For_REX21 example_REX22 ,_, suppose_VV0 we_PPIS2 find_VV0 &lsqb;_( formula_NN1 &rsqb;_) Then_RT we_PPIS2 may_VM set_VVI up_RP the_AT regression_NN1 formula_NN1 &lsqb;_( formula_NN1 &rsqb;_) and_CC if_CS we_PPIS2 begin_VV0 with_IW &lsqb;_( formula_NN1 &rsqb;_) ,_, we_PPIS2 find_VV0 for_IF successive_JJ values_NN2 of_IO &lsqb;_( formula_NN1 &rsqb;_) &lsqb;_( formula_NN1 &rsqb;_) The_AT periodic_JJ change_NN1 of_IO sign_NN1 ,_, with_IW no_AT settled_JJ ratio_NN1 ,_, indicates_VVZ complex_JJ roots_NN2 ._. 
We_PPIS2 therefore_RR apply_VV0 Equation_NN1 (_( 2.7.6.7_MC )_) to_II the_AT last_MD 4_MC figures_NN2 in_II sequences_NN2 of_IO 3_MC (_( we_PPIS2 assume_VV0 contributions_NN2 from_II the_AT other_JJ roots_NN2 to_TO be_VBI vanishingly_RR small_JJ )_) :_: &lsqb;_( formula_NN1 &rsqb;_) which_DDQ give_VV0 us_PPIO2 &lsqb;_( formula_NN1 &rsqb;_) ._. 
Hence_RR (_( 13_MC )_) has_VHZ the_AT factor_NN1 &lsqb;_( formula_NN1 &rsqb;_) ;_; and_CC we_PPIS2 may_VM write_VVI &lsqb;_( formula_NN1 &rsqb;_) so_CS21 that_CS22 all_DB he_PPHS1 factors_VVZ of_IO (_( 13_MC )_) are_VBR complex._NNU 2.9.2_MC Deflation_NN1 of_IO a_AT1 matrix_NN1 pencil_NN1 The_AT eigenvalues_NN2 and_CC vectors_NN2 of_IO a_AT1 simple_JJ pencil_NN1 (_( not_XX necessarily_RR symmetric_JJ )_) will_VM satisfy_VVI &lsqb;_( formula_NN1 &rsqb;_) Write_VV0 these_DD2 as_CSA &lsqb;_( formula_NN1 &rsqb;_) then_RT &lsqb;_( formula_NN1 &rsqb;_) are_VBR evidently_RR the_AT eigenvectors_NN2 belonging_VVG to_II the_AT eigenvalue_NN1 &lsqb;_( formula_NN1 &rsqb;_) i_MC1 of_IO C-1B_FO ._. 
If_CS we_PPIS2 normalise_VV0 them_PPHO2 so_CS21 that_CS22 &lsqb;_( formula_NN1 &rsqb;_) then_RT Sylvester_NP1 expansion_NN1 of_IO C-1B_FO (_( see_VV0 1.18_MC )_) is_VBZ &lsqb;_( formula_NN1 &rsqb;_) so_CS21 that_CS22 we_PPIS2 may_VM express_VVI B_ZZ1 in_II31 terms_II32 of_II33 C_NP1 ,_, etc._RA ,_, as_CSA &lsqb;_( formula_NN1 &rsqb;_) Now_RT suppose_VV0 that_CST we_PPIS2 have_VH0 found_VVN &lsqb;_( formula_NN1 &rsqb;_) ._. 
Then_RT we_PPIS2 are_VBR able_JK to_TO evaluate_VVI ,_, on_II the_AT left-hand_JJ side_NN1 ,_, &lsqb;_( formula_NN1 &rsqb;_) and_CC evidently_RR a_AT1 new_JJ pencil_NN1 ,_, which_DDQ has_VHZ all_DB the_AT eigenvalues_NN2 and_CC vectors_NN2 of_IO the_AT original_JJ pencil_NN1 except_CS21 that_CS22 &lsqb;_( formula_NN1 &rsqb;_) 1_MC1 is_VBZ replaced_VVN by_II 0_MC ,_, is_VBZ &lsqb;_( formula_NN1 &rsqb;_) This_DD1 is_VBZ Lancaster_NP1 's_GE deflation_NN1 formula_NN1 ._. 
We_PPIS2 may_VM note_VVI two_MC points_NN2 :_: (_( i_ZZ1 )_) since_CS &lsqb;_( formula_NN1 &rsqb;_) has_VHZ a_AT1 zero_NN1 root_NN1 ,_, must_VM be_VBI singular_JJ ,_, (_( ii_MC )_) it_PPH1 is_VBZ not_XX necessary_JJ to_TO normalise_VVI the_AT vectors_NN2 ;_; we_PPIS2 can_VM write_VVI B1_FO generally_RR as_CSA &lsqb;_( formula_NN1 &rsqb;_) Example_NN1 1_MC1 We_PPIS2 use_VV0 the_AT pencil_NN1 defined_VVN in_II (_( 2.9.1.8_MC )_) and_CC we_PPIS2 assume_VV0 first_MD that_CST we_PPIS2 have_VH0 found_VVN an_AT1 eigenvalue_NN1 &lsqb;_( formula_NN1 &rsqb;_) 1_MC1 =_FO -1_MC and_CC the_AT associated_JJ eigenvector_NN1 &lsqb;_( formula_NN1 &rsqb;_) ._. 
Since_CS the_AT pencil_NN1 is_VBZ symmetric_JJ we_PPIS2 shall_VM have_VHI y_ZZ1 =_FO x_ZZ1 ._. 
Now_RT ,_, using_VVG (_( 2.9.1.8_MC )_) ,_, we_PPIS2 have_VH0 &lsqb;_( formula_NN1 &rsqb;_) Hence_RR ,_, from_II (_( 5_MC )_) we_PPIS2 find_VV0 &lsqb;_( formula_NN1 &rsqb;_) which_DDQ ,_, with_IW B_ZZ1 defined_VVN as_CSA Diag(1,2,2,1)_FO gives_VVZ &lsqb;_( formula_NN1 &rsqb;_) This_DD1 matrix_NN1 is_VBZ simply_RR singular_JJ ;_; indeed_RR ,_, as_CSA is_VBZ indicated_VVN by_II (_( 3_MC )_) ,_, it_PPH1 is_VBZ annihilated_VVN by_II postmultiplication_NN1 by_II &lsqb;_( formula_NN1 &rsqb;_) :_: in_II (_( 3_MC )_) ,_, &lsqb;_( formula_NN1 &rsqb;_) is_VBZ orthogonal_JJ to_II &lsqb;_( formula_NN1 &rsqb;_) ,_, i_ZZ1 =_FO 2,3_MC ,_, ..._... ,_, n_ZZ1 ._. 
Accordingly_RR ,_, the_AT pencil_NN1 P1_FO (_( &lsqb;_( formula_NN1 &rsqb;_) )_) ;_; has_VHZ a_AT1 zero_NN1 eigenvalue_NN1 ._. 
The_AT deflation_NN1 can_VM of_RR21 course_RR22 be_VBI repeated_VVN ._. 
If_CS ,_, using_VVG P1_FO (_( &lsqb;_( formula_NN1 &rsqb;_) )_) ,_, ;_; we_PPIS2 now_RT find_VV0 an_AT1 eigenvalue_NN1 &lsqb;_( formula_NN1 &rsqb;_) 2_MC =_FO 0.6_MC and_CC the_AT associated_JJ eigenvector_NN1 &lsqb;_( formula_NN1 &rsqb;_) ,_, then_RT &lsqb;_( formula_NN1 &rsqb;_) and_CC then_RT we_PPIS2 may_VM calculate_VVI &lsqb;_( formula_NN1 &rsqb;_) which_DDQ ,_, with_IW B1_FO given_VVN by_II (_( 6_MC )_) ,_, yields_NN2 &lsqb;_( formula_NN1 &rsqb;_) By_II inspection_NN1 ,_, B2_FO is_VBZ doubly_RR degenerate_JJ ;_; it_PPH1 is_VBZ of_RR21 course_RR22 annihilated_VVN by_II postmultiplication_NN1 by_II &lsqb;_( formula_NN1 &rsqb;_) or_CC &lsqb;_( formula_NN1 &rsqb;_) ._. 
A_AT1 third_MD deflation_NN1 would_VM produce_VVI B3_FO ,_, a_AT1 unit_NN1 rank_NN1 matrix_NN1 ,_, which_DDQ in_II this_DD1 instance_NN1 (_( n_ZZ1 =_FO 4_MC )_) would_VM be_VBI the_AT last_MD term_NN1 in_II (_( 3_MC )_) ._. 
2.9.3_MC Iteration_NN1 with_IW submatrices_NN2 of_IO vectors_NN2 We_PPIS2 conclude_VV0 this_DD1 section_NN1 with_IW a_AT1 description_NN1 of_IO a_AT1 method_NN1 often_RR used_VVN in_II the_AT following_JJ circumstances_NN2 :_: (_( a_ZZ1 )_) The_AT problem_NN1 is_VBZ formulated_VVN as_II a_AT1 matrix_NN1 pencil_NN1 ;_; if_CS it_PPH1 is_VBZ of_IO dynamical_JJ origin_NN1 ,_, we_PPIS2 re_II given_JJ a_AT1 non-singular_JJ symmetric_JJ stiffness_NN1 matrix_NN1 C_ZZ1 of_IO order_NN1 n_ZZ1 and_CC a_AT1 corresponding_JJ symmetric_JJ mass_JJ matrix_NN1 B_ZZ1 :_: the_AT problem_NN1 is_VBZ to_TO find_VVI certain_JJ modes_NN2 and_CC frequencies_NN2 satisfying_JJ &lsqb;_( formula_NN1 &rsqb;_) ,_, x_II being_VBG proportional_JJ to_TO exp&lsqb;formula&rsqb;_VVI (_( b_ZZ1 )_) the_AT order_NN1 n_ZZ1 is_VBZ large_JJ ,_, say_VV0 &lsqb;_( formula_NN1 &rsqb;_) (_( c_ZZ1 )_) Interest_NN1 attaches_VVZ only_RR to_II a_AT1 relatively_RR small_JJ number_NN1 ,_, say_VV0 &lsqb;_( formula_NN1 &rsqb;_) ,_, of_IO consecutive_JJ frequencies_NN2 and_CC modes_NN2 ,_, beginning_VVG with_IW the_AT fundamental_JJ ._. 
The_AT process_NN1 is_VBZ one_MC1 of_IO iteration_NN1 with_IW a_AT1 submatrix_NN1 of_IO p_ZZ1 vectors_NN2 ,_, where_CS p_ZZ1 is_VBZ a_RR21 little_RR22 greater_JJR than_CSN the_AT number_NN1 of_IO eigenvalues_NN2 sought_VVD ,_, and_CC involves_VVZ the_AT solution_NN1 at_II each_DD1 step_NN1 of_IO an_AT1 eigenproblem_NN1 of_IO order_NN1 p_ZZ1 only_RR ._. 
Since_CS B_ZZ1 ,_, C_ZZ1 derive_VV0 from_II a_AT1 mechanical_JJ system_NN1 and_CC are_VBR non-singular_JJ ,_, they_PPHS2 will_VM be_VBI pos._NNU def._NNU ,_, and_CC accordingly_RR (_( see_VV0 Theorem_NN1 VIII_MC of_IO 1.22_MC )_) the_AT system_NN1 eigenvalues_NN2 are_VBR all_DB real_JJ ,_, finite_JJ and_CC positive._NNU if_CS we_PPIS2 write_VV0 &lsqb;_( formula_NN1 &rsqb;_) then_RT the_AT full_JJ problem_NN1 may_VM be_VBI written_VVN &lsqb;_( formula_NN1 &rsqb;_) where_RRQ X_ZZ1 is_VBZ the_AT modal_JJ matrix_NN1 ,_, &lsqb;_( formula_NN1 &rsqb;_) the_AT spectral_JJ matrix_NN1 ,_, and_CC A_ZZ1 =_FO C-1B_FO the_AT dynamical_JJ matrix_NN1 ._. 
It_PPH1 is_VBZ assumed_VVN that_CST the_AT eigenvalues_NN2 in_II &lsqb;_( formula_NN1 &rsqb;_) (_( and_CC the_AT corresponding_JJ eigenvectors_NN2 in_II X_ZZ1 )_) are_VBR so_RR ordered_VVN that_CST &lsqb;_( formula_NN1 &rsqb;_) has_VHZ the_AT eigenvalues_NN2 &lsqb;_( formula_NN1 &rsqb;_) in_II descending_JJ order_NN1 of_IO magnitude_NN1 down_II the_AT diagonal_JJ ,_, so_CS21 that_CS22 &lsqb;_( formula_NN1 &rsqb;_) 1_MC1 and_CC the_AT first_MD vector_NN1 in_II X_ZZ1 define_VV0 the_AT fundamental_JJ frequency_NN1 and_CC mode_NN1 ._. 
We_PPIS2 are_VBR not_XX concerned_JJ to_TO solve_VVI (_( 1_MC1 )_) as_II a_AT1 whole_NN1 ._. 
If_CS it_PPH1 is_VBZ partitioned_VVN in_II the_AT form_NN1 &lsqb;_( formula_NN1 &rsqb;_) where_RRQ Y_ZZ1 is_VBZ of_IO order_NN1 (_( n_ZZ1 p_ZZ1 )_) and_CC &lsqb;_( formula_NN1 &rsqb;_) square_NN1 ,_, of_IO order_NN1 p_ZZ1 ,_, (_( 2_MC )_) gives_VVZ &lsqb;_( formula_NN1 &rsqb;_) and_CC only_RR the_AT first_MD of_IO these_DD2 equations_NN2 concerns_VVZ us_PPIO2 ._. 
However_RR ,_, we_PPIS2 may_VM note_VVI that_CST (_( see_VV0 1.19_MC )_) premultiplication_NN1 by_II &lsqb;_( formula_NN1 &rsqb;_) of_IO the_AT first_MD equation_NN1 in_II (_( 1_MC1 )_) ,_, and_CC transposition_NN1 ,_, shows_VVZ that_CST &lsqb;_( formula_NN1 &rsqb;_) are_VBR both_RR diagonal_JJ ._. 
For_IF analytical_JJ purposes_NN2 we_PPIS2 may_VM normalise_VVI X_ZZ1 so_CS21 that_CS22 &lsqb;_( formula_NN1 &rsqb;_) which_DDQ imply_VV0 ,_, inter_RR21 alia_RR22 ,_, &lsqb;_( formula_NN1 &rsqb;_) Having_VHG set_VVN out_RP some_DD preliminary_JJ considerations_NN2 ,_, we_PPIS2 now_RT state_VV0 the_AT method_NN1 ._. 
Select_VV0 a_AT1 set_NN1 of_IO starting_VVG vectors_NN2 Yo_UH ,_, arbitrary_JJ except_CS21 that_CS22 they_PPHS2 must_VM be_VBI linearly_RR independent_JJ ;_; then_RT evaluate_VV0 the_AT (_( n_ZZ1 p_ZZ1 )_) matrix_NN1 &lsqb;_( formula_NN1 &rsqb;_) and_CC use_VV0 Wo_NP1 to_TO form_VVI the_AT two_MC square_JJ matrices_NN2 of_IO order_NN1 p_ZZ1 &lsqb;_( formula_NN1 &rsqb;_) Now_RT solve_VV0 the_AT eigenproblem_NN1 (_( any_DD suitable_JJ method_NN1 from_II this_DD1 chapter_NN1 may_VM be_VBI used_VVN )_) &lsqb;_( formula_NN1 &rsqb;_) for_IF its_APPGE modal_JJ matrix_NN1 Mo_NP1 ;_; the_AT spectral_JJ matrix_NN1 Ao_NP1 which_DDQ emerges_VVZ is_VBZ (_( see_VV0 below_RL )_) a_AT1 first_MD approximation_NN1 to_II Ay_UH ._. 
Then_RT an_AT1 improved_JJ approximation_NN1 to_II Y_ZZ1 is_VBZ &lsqb;_( formula_NN1 &rsqb;_) where_RRQ do_VD0 is_VBZ an_AT1 arbitrary_JJ non-singular_JJ diagonal_JJ matrix_NN1 which_DDQ may_VM ,_, for_REX21 example_REX22 ,_, be_VBI used_VVN to_TO make_VVI a_AT1 homologous_JJ element_NN1 in_II each_DD1 of_IO the_AT corresponding_JJ columns_NN2 of_IO Y1_FO and_CC Yo_UH the_AT same_DA ;_; do_VD0 is_VBZ useful_JJ ,_, but_CCB not_XX essential_JJ ._. 
This_DD1 completes_VVZ one_MC1 step_NN1 ._. 
The_AT next_MD step_NN1 uses_VVZ Y1_FO in_II (_( 6_MC )_) to_TO give_VVI W1_FO =_FO AY1_FO ,_, and_CC the_AT cycle_NN1 of_IO operations_NN2 is_VBZ repeated_VVN until_II convergence_NN1 occurs_VVZ ._. 
We_PPIS2 may_VM note_VVI some_DD interesting_JJ aspects_NN2 of_IO the_AT method._NNU (_( i_ZZ1 )_) If_CS Yo_UH consists_VVZ of_IO one_MC1 vector_NN1 only_RR ,_, so_RR does_VDZ Wo_NP1 ,_, and_CC then_RT &lsqb;_( formula_NN1 &rsqb;_) are_VBR scalars_NN2 ,_, s_ZZ1 is_VBZ the_AT modal_JJ matrix_NN1 &lsqb;_( formula_NN1 &rsqb;_) ._. 
If_CS we_PPIS2 choose_VV0 &lsqb;_( formula_NN1 &rsqb;_) in_II (_( 9_MC )_) then_RT &lsqb;_( formula_NN1 &rsqb;_) and_CC this_DD1 is_VBZ the_AT single_JJ vector_NN1 iteration_NN1 discussed_VVN in_II 2.7.1_MC ._. 
Moreover_RR ,_, the_AT trivial_JJ eigenproblem_NN1 gives_VVZ a_AT1 first_MD approximation_NN1 to_II &lsqb;_( formula_NN1 &rsqb;_) 1_MC1 as_II the_AT Rayleigh_NP1 (_( ii_MC )_) If_CS Yo_UH happens_VVZ to_TO be_VBI the_AT correct_JJ set_NN1 Y_ZZ1 ,_, the_AT method_NN1 makes_VVZ this_DD1 apparent_JJ at_RR21 once_RR22 ,_, for_IF then_RT in_II31 view_II32 of_II33 (_( 3_MC )_) &lsqb;_( formula_NN1 &rsqb;_) so_CS21 that_CS22 ,_, on_II use_NN1 of_IO (_( 5_MC )_) &lsqb;_( formula_NN1 &rsqb;_) Thus_RR both_DB2 matrices_NN2 are_VBR diagonal_JJ ,_, so_CS21 that_CS22 the_AT eigenproblem_NN1 is_VBZ again_RT trivial_JJ ,_, having_VHG the_AT solution_NN1 &lsqb;_( formula_NN1 &rsqb;_) ._. 
Then_RT &lsqb;_( formula_NN1 &rsqb;_) when_CS scaled_VVN appropriately_RR ,_, repeats_NN2 &lsqb;_( formula_NN1 &rsqb;_) showing_VVG that_CST both_DB2 equal_JJ Y._NP1 (_( iii_MC )_) If_CS Yo_UH happens_VVZ to_TO be_VBI a_AT1 set_NN1 of_IO linear_JJ combinations_NN2 of_IO the_AT vectors_NN2 in_II Y_ZZ1 ,_, the_AT correct_JJ solution_NN1 is_VBZ obtained_VVN in_II one_MC1 step_NN1 ._. 
For_CS we_PPIS2 may_VM then_RT write_VVI &lsqb;_( formula_NN1 &rsqb;_) where_RRQ P_ZZ1 is_VBZ a_AT1 square_JJ matrix_NN1 of_IO order_NN1 p_ZZ1 ,_, non-singular_JJ since_CS the_AT columns_NN2 of_IO Yo_UH and_CC of_IO Y_ZZ1 ,_, separately_RR ,_, are_VBR linearly_RR independent_JJ ._. 
Then_RT &lsqb;_( formula_NN1 &rsqb;_) in_II31 view_II32 of_II33 (_( 3_MC )_) ._. 
Thus_RR ,_, on_II use_NN1 of_IO (_( 5_MC )_) ,_, &lsqb;_( formula_NN1 &rsqb;_) On_II premultiplication_NN1 by_II &lsqb;_( formula_NN1 &rsqb;_) ,_, the_AT eigenproblem_NN1 (_( 8_MC )_) reduces_VVZ to_II &lsqb;_( formula_NN1 &rsqb;_) which_DDQ has_VHZ the_AT solution_NN1 &lsqb;_( formula_NN1 &rsqb;_) If_CS do_VD0 is_VBZ chosen_VVN as_CSA &lsqb;_( formula_NN1 &rsqb;_) The_AT condition_NN1 envisaged_VVN in_II (_( iii_MC )_) is_VBZ ,_, in_RR21 general_RR22 ,_, unlikely_JJ ,_, unless_CS for_REX21 example_REX22 a_AT1 neighbour_NN1 system_NN1 (_( see_VV0 Chapter_NN1 3_MC )_) supplies_VVZ a_AT1 good_JJ approximation_NN1 to_II Y._NP1 In_II most_DAT cases_NN2 ,_, we_PPIS2 must_VM write_VVI &lsqb;_( formula_NN1 &rsqb;_) where_RRQ R_ZZ1 ,_, like_II Yo_UH ,_, is_VBZ of_IO order_NN1 (_( n_ZZ1 p_ZZ1 )_) ._. 
In_II this_DD1 general_JJ case_NN1 &lsqb;_( formula_NN1 &rsqb;_) in_II31 view_II32 of_II33 (_( 1_MC1 )_) ._. 
Then_RT the_AT eigenproblem_NN1 matrices_NN2 become_VV0 &lsqb;_( formula_NN1 &rsqb;_) where_RRQ (_( 4_MC )_) has_VHZ been_VBN used_VVN ._. 
We_PPIS2 therefore_RR have_VH0 to_TO solve_VVI for_IF &lsqb;_( formula_NN1 &rsqb;_) &lsqb;_( formula_NN1 &rsqb;_) Knowing_VVG Mo_NP1 ,_, we_PPIS2 now_RT use_VV0 (_( 11_MC )_) to_TO evaluate_VVI &lsqb;_( formula_NN1 &rsqb;_) whence_RRQ in_II turn_NN1 we_PPIS2 find_VV0 &lsqb;_( formula_NN1 &rsqb;_) Since_CS &lsqb;_( formula_NN1 &rsqb;_) is_VBZ non-singular_JJ ,_, we_PPIS2 may_VM reduce_VVI the_AT eigenproblem_NN1 based_VVN on_II W1_FO to_II &lsqb;_( formula_NN1 &rsqb;_) which_DDQ we_PPIS2 solve_VV0 for_IF the_AT unknown_JJ &lsqb;_( formula_NN1 &rsqb;_) ._. 
A_AT1 third_MD step_NN1 leads_VVZ to_II &lsqb;_( formula_NN1 &rsqb;_) It_PPH1 is_VBZ clear_JJ from_II (_( 13_MC )_) ,_, (_( 16_MC )_) and_CC (_( 17_MC )_) that_CST after_CS r_ZZ1 steps_VVZ we_PPIS2 have_VH0 to_TO solve_VVI &lsqb;_( formula_NN1 &rsqb;_) If_CS &lsqb;_( formula_NN1 &rsqb;_) is_VBZ written_VVN as_CSA M_ZZ1 ,_, (_( 18_MC )_) in_II partitioned_JJ form_NN1 is_VBZ (_( see_VV0 (_( 10_MC )_) )_) &lsqb;_( formula_NN1 &rsqb;_) or_CC ,_, on_II expansion_NN1 ,_, &lsqb;_( formula_NN1 &rsqb;_) Now_RT ,_, since_CS all_DB the_AT eigenvalues_NN2 in_II &lsqb;_( formula_NN1 &rsqb;_) are_VBR greater_JJR than_CSN any_DD of_IO those_DD2 in_II &lsqb;_( formula_NN1 &rsqb;_) ,_, when_CS r_ZZ1 is_VBZ sufficiently_RR large_JJ the_AT terms_NN2 in_II &lsqb;_( formula_NN1 &rsqb;_) become_VV0 negligible_JJ compared_VVN with_IW those_DD2 in_II &lsqb;_( formula_NN1 &rsqb;_) ,_, when_RRQ (_( 19_MC )_) reduces_VVZ to_II &lsqb;_( formula_NN1 &rsqb;_) and_CC if_CS s_ZZ1 is_VBZ on-singular_JJ ,_, this_DD1 in_II turn_NN1 reduces_VVZ to_II &lsqb;_( formula_NN1 &rsqb;_) which_DDQ has_VHZ the_AT solution_NN1 &lsqb;_( formula_NN1 &rsqb;_) ._. 
This_DD1 shows_VVZ that_DD1 convergence_NN1 is_VBZ complete_JJ ._. 
Two_MC things_NN2 must_VM be_VBI noted_VVN here_RL ._. 
First_MD ,_, although_CS when_CS r_ZZ1 is_VBZ large_JJ ,_, in_II an_AT1 overall_JJ sense_NN1 the_AT terms_NN2 in_II &lsqb;_( formula_NN1 &rsqb;_) those_DD2 in_II &lsqb;_( formula_NN1 &rsqb;_) ,_, yet_RR the_AT smallest_JJT element_NN1 in_II &lsqb;_( formula_NN1 &rsqb;_) is_VBZ not_XX necessarily_RR much_RR larger_JJR than_CSN the_AT largest_JJT element_NN1 in_II &lsqb;_( formula_NN1 &rsqb;_) ._. 
As_II a_AT1 result_NN1 calculation_NN1 of_IO the_AT last_MD two_MC or_CC three_MC eigenvalues_NN2 in_II &lsqb;_( formula_NN1 &rsqb;_) may_VM not_XX be_VBI very_RG accurate_JJ ._. 
It_PPH1 is_VBZ for_IF this_DD1 reason_NN1 that_CST we_PPIS2 choose_VV0 p_ZZ1 to_TO be_VBI a_RR21 little_RR22 greater_JJR than_CSN the_AT number_NN1 of_IO eigenvalues_NN2 sought_VVD ._. 
Second_MD ,_, reduction_NN1 of_IO the_AT eigenproblem_NN1 to_II (_( 20_MC )_) requires_VVZ s_ZZ1 to_TO be_VBI non-singular_JJ ._. 
Now_RT R_ZZ1 has_VHZ p_NNU linearly_RR independent_JJ columns_NN2 ,_, and_CC therefore_RR also_RR p_ZZ1 linearly_RR independent_JJ rows_NN2 ._. 
Thus_RR s_ZZ1 can_VM be_VBI non-singular_JJ ;_; but_CCB it_PPH1 is_VBZ not_XX necessarily_RR so_RR ,_, and_CC then_RT Y_ZZ1 as_CSA given_VVN by_II the_AT method_NN1 may_VM be_VBI deficient_JJ ._. 
For_REX21 example_REX22 ,_, if_CS the_AT top_JJ row_NN1 of_IO s_ZZ1 were_VBDR null_JJ ,_, the_AT first_MD vector_NN1 in_II X_ZZ1 (_( the_AT fundamental_JJ mode_NN1 )_) would_VM be_VBI absent_JJ in_II the_AT make-up_NN1 of_IO Yo_UH ,_, and_CC this_DD1 absence_NN1 would_VM persist_VVI ._. 
There_EX is_VBZ no_AT certain_JJ way_NN1 of_IO avoiding_VVG this_DD1 difficulty_NN1 when_CS Yo_UH is_VBZ chosen_VVN arbitrarily_RR ;_; but_CCB in_II practice_NN1 it_PPH1 does_VDZ not_XX often_RR arise_VVI ._. 
Although_CS the_AT eigenproblem_NN1 to_TO be_VBI solved_VVN at_II each_DD1 step_NN1 is_VBZ vastly_RR less_RGR time-consuming_JJ and_CC expensive_JJ than_CSN direct_JJ methods_NN2 applied_VVN to_II the_AT full_JJ system_NN1 ,_, yet_RR it_PPH1 does_VDZ take_VVI much_DA1 time_NNT1 ._. 
However_RR ,_, after_CS a_AT1 few_DA2 steps_NN2 v_ZZ1 ,_, and_CC &lsqb;_( formula_NN1 &rsqb;_) usually_RR become_VV0 heavily_RR diagonal_JJ ,_, so_CS21 that_CS22 Mr_NNB tends_VVZ towards_II I_MC1 ;_; this_DD1 occurs_VVZ before_II the_AT vectors_NN2 Yr_NNU have_VH0 converged_VVN ._. 
In_II these_DD2 circumstances_NN2 ,_, use_NN1 of_IO the_AT Collar-Jahn_NP1 method_NN1 ,_, described_VVN in_II Chapter_NN1 3_MC ;_; may_VM greatly_RR expedite_VVI the_AT solution_NN1 ._. 
An_AT1 example_NN1 of_IO this_DD1 is_VBZ given_VVN in_II 3.9_MC ._. 
Further_JJR reference_NN1 to_II the_AT method_NN1 is_VBZ also_RR made_VVN in_II 8.5_MC ._. 
Example_NN1 1_MC1 The_AT small-order_JJ example_NN1 which_DDQ is_VBZ all_DB we_PPIS2 can_VM give_VVI here_RL does_VDZ not_XX of_RR21 course_RR22 do_VDI justice_NN1 to_II the_AT method_NN1 ,_, but_CCB at_RR21 least_RR22 shows_VVZ how_RRQ it_PPH1 works_VVZ ._. 
We_PPIS2 begin_VV0 with_IW an_AT1 illustration_NN1 of_IO the_AT situation_NN1 in_II (_( iii_MC )_) above_RL ;_; the_AT reader_NN1 is_VBZ invited_VVN to_TO check_VVI the_AT arithmetic_NN1 ._. 
Let_VV0 &lsqb;_( formula_NN1 &rsqb;_) Then_RT &lsqb;_( formula_NN1 &rsqb;_) Suppose_VV0 now_RT we_PPIS2 are_VBR given_VVN ,_, or_CC choose_VV0 ,_, &lsqb;_( formula_NN1 &rsqb;_) It_PPH1 then_RT follows_VVZ that_CST &lsqb;_( formula_NN1 &rsqb;_) The_AT eigenvalues_NN2 of_IO &lsqb;_( formula_NN1 &rsqb;_) are_VBR thus_RR 10_MC and_CC 5_MC and_CC &lsqb;_( formula_NN1 &rsqb;_) The_AT reader_NN1 should_VM check_VVI that_CST these_DD2 are_VBR eigenvectors_NN2 of_IO A_ZZ1 and_CC that_CST they_PPHS2 correspond_VV0 to_II eigenvalues_NN2 10_MC and_CC 5_MC ._. 
We_PPIS2 have_VH0 thus_RR obtained_VVN our_APPGE solution_NN1 ,_, in_II this_DD1 case_NN1 ,_, in_II one_MC1 step_NN1 ._. 
It_PPH1 will_VM be_VBI found_VVN that_CST each_DD1 column_NN1 of_IO &lsqb;_( formula_NN1 &rsqb;_) is_VBZ in_II fact_NN1 ,_, a_AT1 linear_JJ combination_NN1 of_IO the_AT columns_NN2 of_IO &lsqb;_( formula_NN1 &rsqb;_) Example_NN1 2_MC We_PPIS2 use_VV0 the_AT same_DA A_ZZ1 ,_, B_ZZ1 ,_, C_ZZ1 ,_, as_CSA in_II Example_NN1 1_MC1 ,_, but_CCB select_VV0 as_CSA our_APPGE starting_NN1 vector_NN1 submatrix_NN1 &lsqb;_( formula_NN1 &rsqb;_) In_II choosing_VVG Yo_UH we_PPIS2 have_VH0 selected_VVN simple_JJ vectors_NN2 which_DDQ are_VBR clearly_RR linearly_RR independent_JJ ,_, and_CC which_DDQ between_II them_PPHO2 bring_VV0 all_DB elements_NN2 in_II A_ZZ1 into_II operation_NN1 in_II the_AT formation_NN1 of_IO Wo_NP1 ._. 
We_PPIS2 then_RT find_VV0 &lsqb;_( formula_NN1 &rsqb;_) where_RRQ the_AT product_NN1 is_VBZ quoted_VVN to_II four_MC significant_JJ figures_NN2 ._. 
Its_APPGE eigenvalues_NN2 are_VBR 9.805_MC ,_, 2.411_MC and_CC ,_, very_RG approximately_RR (_( great_JJ accuracy_NN1 is_VBZ not_XX necessary_JJ )_) ,_, &lsqb;_( formula_NN1 &rsqb;_) which_DDQ yields_VVZ ,_, again_RT approximately_RR ,_, &lsqb;_( formula_NN1 &rsqb;_) Proceeding_VVG in_II this_DD1 way_NN1 ,_, we_PPIS2 find_VV0 at_II the_AT end_NN1 of_IO the_AT fifth_MD step_NN1 &lsqb;_( formula_NN1 &rsqb;_) and_CC in_II the_AT sixth_MD step_VV0 this_DD1 gives_VVZ &lsqb;_( formula_NN1 &rsqb;_) These_DD2 are_VBR so_RG nearly_RR diagonal_JJ that_CST the_AT eigenproblem_NN1 is_VBZ trivial_JJ ;_; the_AT eigenvalues_NN2 are_VBR 10_MC ,_, 4.9987_MC ._. 
However_RR ,_, we_PPIS2 do_VD0 not_XX obtain_VVI convergence_NN1 until_CS after_CS the_AT 10th_MD step_NN1 ,_, which_DDQ gives_VVZ eigenvalues_NN2 10_MC ,_, 5_MC and_CC &lsqb;_( formula_NN1 &rsqb;_) The_AT second_MD columns_NN2 should_VM be_VBI 1,1,0_MC ,_, -1_MC ._. 
2.9.4_MC A_ZZ1 variant_NN1 We_PPIS2 now_RT give_VV0 a_AT1 variant_NN1 of_IO the_AT method_NN1 of_IO 2.9.3_MC ;_; we_PPIS2 do_VD0 so_RR partly_RR because_CS the_AT variant_NN1 can_VM save_VVI computer_NN1 time_NNT1 ,_, especially_RR if_CS may_VM iterations_NN2 are_VBR required_VVN ,_, but_CCB also_RR because_CS it_PPH1 enables_VVZ us_PPIO2 simply_RR to_TO bring_VVI out_RP certain_JJ features_NN2 of_IO the_AT method_NN1 ,_, which_DDQ lead_VV0 to_II possible_JJ modifications_NN2 ._. 
For_IF simplicity_NN1 ,_, we_PPIS2 assume_VV0 that_CST the_AT eigenvalues_NN2 sought_VVN are_VBR all_DB different_JJ ._. 
We_PPIS2 are_VBR given_VVN the_AT matrices_NN2 C_ZZ1 and_CC B_ZZ1 ,_, both_RR symmetric_JJ an_AT1 pos._NNU def_NN1 ._. 
We_PPIS2 may_VM therefore_RR use_VVI Choleski_NP1 's_GE method_NN1 (_( see_VV0 2.4.2_MC )_) to_TO find_VVI a_AT1 lower_JJR triangular_JJ matrix_NN1 L_ZZ1 such_CS21 that_CS22 &lsqb;_( formula_NN1 &rsqb;_) ,_, and_CC may_VM then_RT evaluate_VVI &lsqb;_( formula_NN1 &rsqb;_) ,_, accordingly_RR the_AT equation_NN1 &lsqb;_( formula_NN1 &rsqb;_) may_VM be_VBI written_VVN &lsqb;_( formula_NN1 &rsqb;_) and_CC S_ZZ1 is_VBZ symmetric_JJ and_CC pos._NNU def_NN1 ._. 
Now_RT ,_, instead_II21 of_II22 using_VVG the_AT three_MC matrices_NN2 C_ZZ1 ,_, B_ZZ1 and_CC A_ZZ1 =_FO C-1B_FO as_CSA in_II 2.9.3_MC we_PPIS2 work_VV0 only_RR with_IW S._NP1 Suppose_VV0 we_PPIS2 now_RT proceed_VV0 as_CSA in_II 2.9.3_MC :_: we_PPIS2 begin_VV0 with_IW the_AT submatrix_NN1 Yo_UH and_CC successively_RR evaluate_VV0 &lsqb;_( formula_NN1 &rsqb;_) and_CC we_PPIS2 then_RT solve_VV0 for_IF Mo_NP1 the_AT eigenproblem_NN1 &lsqb;_( formula_NN1 &rsqb;_) Then_RT a_AT1 new_JJ approximation_NN1 to_II Y_ZZ1 is_VBZ &lsqb;_( formula_NN1 &rsqb;_) At_II this_DD1 point_NN1 we_PPIS2 may_VM ask_VVI :_: what_DDQ is_VBZ the_AT purpose_NN1 of_IO eigenproblem_NN1 ?_? 
The_AT answer_NN1 appears_VVZ if_CS we_PPIS2 premultiply_RR and_CC postmultiply_JJ Equation_NN1 (_( 5_MC )_) by_II appropriate_JJ quantities_NN2 to_TO give_VVI since_RR do_VD0 permutes_NN2 with_IW &lsqb;_( formula_NN1 &rsqb;_) ,_, &lsqb;_( formula_NN1 &rsqb;_) on_II use_NN1 of_IO (_( 6_MC )_) ._. 
Transposition_NN1 of_IO (_( 7_MC )_) shows_NN2 (_( compare_VV0 1.19_MC )_) that_CST &lsqb;_( formula_NN1 &rsqb;_) permutes_VVZ with_IW &lsqb;_( formula_NN1 &rsqb;_) and_CC is_VBZ therefore_RR diagonal_JJ ,_, as_CSA is_VBZ &lsqb;_( formula_NN1 &rsqb;_) ._. 
The_AT columns_NN2 of_IO &lsqb;_( formula_NN1 &rsqb;_) are_VBR thus_RR mutually_RR orthogonal_JJ ;_; this_DD1 is_VBZ the_AT purpose_NN1 of_IO evaluating_VVG Mo_NP1 :_: its_APPGE use_NN1 orthogonalises_VVZ the_AT columns_NN2 ,_, an_AT1 our_APPGE ultimate_JJ objective_NN1 is_VBZ the_AT orthogonal_JJ set_NN1 Y._NP1 A_ZZ1 second_NNT1 step_NN1 with_IW Yo_UH in_II (_( 2_MC )_) replaced_VVD by_II Y1_FO ,_, leads_VVZ to_II the_AT eigenproblem_NN1 (_( compare_VV0 (_( 5_MC )_) )_) &lsqb;_( formula_NN1 &rsqb;_) Again_RT we_PPIS2 may_VM pause_VVI and_CC observe_VVI that_CST (_( 8_MC )_) gives_VVZ us_PPIO2 &lsqb;_( formula_NN1 &rsqb;_) directly_RR ;_; we_PPIS2 do_VD0 not_XX need_VVI to_TO find_VVI Mo_NP1 first_MD from_II (_( 5_MC )_) and_CC then_RT M1_FO from_II (_( 8_MC )_) ._. 
Thus_RR the_AT step_NN1 (_( 5_MC )_) is_VBZ unnecessary_JJ ._. 
Then_RT ,_, in_II its_APPGE turn_NN1 ,_, the_AT step_NN1 (_( 8_MC )_) is_VBZ unnecessary_JJ ;_; and_RR31 so_RR32 on_RR33 ._. 
Indeed_RR ,_, in_II theory_NN1 all_DB that_DD1 is_VBZ required_VVN is_VBZ that_CST a_AT1 sufficient_JJ number_NN1 of_IO direct_JJ iterations_NN2 is_VBZ made_VVN ,_, with_IW the_AT columns_NN2 orthogonalised_VVN as_II the_AT last_MD step_NN1 ;_; in_II practice_NN1 ,_, however_RR ,_, the_AT numbers_NN2 would_VM become_VVI increasingly_RR ill-conditioned_JJ as_CSA S_ZZ1 is_VBZ raised_VVN to_II a_AT1 high_JJ power_NN1 and_CC approximates_VVZ to_II a_AT1 unit_NN1 rank_NN1 matrix_NN1 ._. 
However_RR ,_, it_PPH1 is_VBZ probably_RR sufficient_JJ if_CS the_AT columns_NN2 are_VBR orthogonalised_VVN at_II ,_, say_VV0 ,_, every_AT1 fourth_MD iteration_NN1 ._. 
If_CS we_PPIS2 do_VD0 this_DD1 ,_, the_AT procedure_NN1 is_VBZ as_CSA follows_VVZ ._. 
Begin_VV0 with_IW Yo_UH and_CC evaluate_VV0 successively_RR &lsqb;_( formula_NN1 &rsqb;_) ._. 
At_II this_DD1 stage_NN1 ,_, evaluate_VV0 &lsqb;_( formula_NN1 &rsqb;_) and_CC solve_VV0 the_AT eigenproblem_NN1 &lsqb;_( formula_NN1 &rsqb;_) for_IF M._NN1 Then_RT a_AT1 new_JJ approximation_NN1 ,_, which_DDQ has_VHZ mutually_RR orthogonal_JJ columns_NN2 ,_, is_VBZ &lsqb;_( formula_NN1 &rsqb;_) ,_, with_IW which_DDQ in_II31 place_II32 of_II33 Yo_UH the_AT cycle_NN1 is_VBZ repeated_VVN ,_, and_RR31 so_RR32 on_RR33 ._. 
Example_NN1 1_MC1 We_PPIS2 use_VV0 the_AT same_DA matrices_NN2 B_ZZ1 and_CC C_ZZ1 as_CSA in_II Example_NN1 2.9.3.1._MC then_RT ,_, if_CS &lsqb;_( formula_NN1 &rsqb;_) ,_, &lsqb;_( formula_NN1 &rsqb;_) ,_, we_PPIS2 find_VV0 ,_, to_II four_MC decimal_JJ places_NN2 ,_, &lsqb;_( formula_NN1 &rsqb;_) &lsqb;_( formula_NN1 &rsqb;_) As_CSA our_APPGE starting_NN1 matrix_NN1 we_PPIS2 choose_VV0 &lsqb;_( formula_NN1 &rsqb;_) This_DD1 has_VHZ mutually_RR orthogonal_JJ columns_NN2 ;_; however_RR ,_, as_CSA we_PPIS2 shall_VM see_VVI later_RRR ,_, it_PPH1 is_VBZ not_XX a_AT1 good_JJ initial_JJ choice_NN1 for_IF the_AT first_MD column_NN1 ._. 
We_PPIS2 ow_VV0 form_NN1 &lsqb;_( formula_NN1 &rsqb;_) ,_, at_II each_DD1 stage_NN1 making_VVG the_AT &quot;_" 11_MC &quot;_" and_CC &quot;_" 22_MC &quot;_" elements_NN2 unity_NN1 ._. 
We_PPIS2 find_VV0 &lsqb;_( formula_NN1 &rsqb;_) we_PPIS2 require_VV0 &lsqb;_( formula_NN1 &rsqb;_) as_CSA it_PPH1 stands_VVZ ._. 
We_PPIS2 now_RT form_VV0 v_ZZ1 ,_, &lsqb;_( formula_NN1 &rsqb;_) as_CSA in_II (_( 9_MC )_) ,_, (_( 10_MC )_) ,_, and_CC solve_VV0 (_( 11_MC )_) for_IF M._NN1 To_II sufficient_JJ accuracy_NN1 ,_, we_PPIS2 find_VV0 &lsqb;_( formula_NN1 &rsqb;_) and_CC then_RT a_AT1 new_JJ Yo_UH is_VBZ ,_, with_IW d_ZZ1 chosen_VVN appropriately_RR ,_, &lsqb;_( formula_NN1 &rsqb;_) and_CC it_PPH1 may_VM be_VBI checked_VVN that_CST these_DD2 two_MC columns_NN2 are_VBR ,_, sufficiently_RR nearly_RR ,_, orthogonal_JJ ._. 
We_PPIS2 now_RT repeat_VV0 the_AT cycle_NN1 ._. 
This_DD1 time_NNT1 we_PPIS2 find_VV0 &lsqb;_( formula_NN1 &rsqb;_) and_CC &lsqb;_( formula_NN1 &rsqb;_) and_CC again_RT ,_, sufficiently_RR nearly_RR ,_, the_AT columns_NN2 are_VBR orthogonal_JJ ._. 
A_AT1 third_MD cycle_NN1 leads_VVZ to_II the_AT eigenvectors_NN2 (_( and_CC eigenvalues_NN2 )_) of_IO S_ZZ1 :_: &lsqb;_( formula_NN1 &rsqb;_) We_PPIS2 leave_VV0 it_PPH1 to_II the_AT reader_NN1 to_TO establish_VVI that_CST the_AT eigenvectors_NN2 of_IO the_AT original_JJ system_NN1 ,_, given_VVN by_II premultiplication_NN1 of_IO w_ZZ1 by_II &lsqb;_( formula_NN1 &rsqb;_) may_VM be_VBI written_VVN &lsqb;_( formula_NN1 &rsqb;_) 2.10_MC IMPROVEMENT_NN1 OF_IO THE_AT ACCURACY_NN1 OF_IO EIGENVALUES_NN2 AND_CC VECTORS_NN2 It_PPH1 frequently_RR happens_VVZ ,_, both_RR in_II dynamical_JJ problems_NN2 and_CC in_II other_JJ eigenvalue_NN1 problems_NN2 that_CST we_PPIS2 have_VH0 approximate_JJ values_NN2 for_IF one_MC1 or_CC more_DAR eigenvalues_NN2 (_( and_CC perhaps_RR vectors_NN2 )_) and_CC require_VV0 to_TO obtain_VVI more_RGR exact_JJ values_NN2 ._. 
This_DD1 problem_NN1 ,_, from_II another_DD1 viewpoint_NN1 ,_, is_VBZ discussed_VVN more_RGR fully_RR in_II Chapter_NN1 3_MC ._. 
Here_RL we_PPIS2 shall_VM deal_VVI only_RR with_IW some_DD simple_JJ cases._NNU 2.10.1_MC Problem_NN1 formulated_VVN as_II a_AT1 matrix_NN1 pencil_NN1 For_IF simplicity_NN1 ,_, let_VV0 us_PPIO2 first_MD discuss_VVI the_AT undamped_JJ oscillations_NN2 of_IO a_AT1 mechanical_JJ system_NN1 having_VHG n_ZZ1 degrees_NN2 of_IO freedom_NN1 ,_, the_AT equations_NN2 of_IO motion_NN1 of_IO which_DDQ have_VH0 been_VBN derived_VVN by_II Lagrangian_JJ methods_NN2 from_II energy_NN1 considerations_NN2 ;_; they_PPHS2 will_VM appear_VVI ,_, in_II the_AT usual_JJ notation_NN1 of_IO dynamics_NN ,_, as_CSA &lsqb;_( formula_NN1 &rsqb;_) Here_RL x_ZZ1 is_VBZ the_AT column_NN1 of_IO coordinates_NN2 ;_; in_II what_DDQ follows_VVZ ,_, we_PPIS2 assume_VV0 the_AT last_MD element_NN1 x_ZZ1 to_TO be_VBI non-nodal_JJ (_( if_CS it_PPH1 is_VBZ not_XX ,_, it_PPH1 can_VM be_VBI made_VVN so_RR by_II rewriting_VVG the_AT equations_NN2 in_II a_AT1 different_JJ order_NN1 )_) ._. 
Also_RR C_ZZ1 ,_, the_AT inertia_NN1 matrix_NN1 ,_, will_VM be_VBI symmetric_JJ and_CC pos._NNU def._NNU and_CC B_ZZ1 ,_, the_AT stiffness_NN1 matrix_NN1 ,_, also_RR symmetric_JJ and_CC non-neg._JJ def_NN1 ;_; finally_RR ,_, F_ZZ1 is_VBZ a_AT1 column_NN1 of_IO applied_JJ forcing_NN1 functions_NN2 ._. 
In_II simple_JJ harmonic_JJ motion_NN1 ,_, these_DD2 forces_NN2 ,_, and_CC x_ZZ1 ,_, will_VM be_VBI proportional_JJ to_II exp_JJ &lsqb;_( formula_NN1 &rsqb;_) and_CC if_CS we_PPIS2 then_RT write_VV0 &lsqb;_( formula_NN1 &rsqb;_) for_IF go_NN1 2_MC ,_, (_( 1_MC1 )_) reduces_VVZ to_II the_AT usual_JJ formula_NN1 &lsqb;_( formula_NN1 &rsqb;_) ._. 
We_PPIS2 observe_VV0 that_CST ,_, in_II31 addition_II32 to_II33 the_AT given_JJ matrices_NN2 C_ZZ1 and_CC B_ZZ1 ,_, (_( 2_MC )_) contains_VVZ 2n_FO +_FO 1_MC1 other_JJ quantities_NN2 ,_, viz._REX ,_, x_ZZ1 ,_, and_CC F._NP1 Since_CS we_PPIS2 have_VH0 n_ZZ1 equations_NN2 ,_, we_PPIS2 can_VM determine_VVI any_DD n_ZZ1 of_IO these_DD2 quantities_NN2 in_II31 terms_II32 of_II33 the_AT remaining_JJ n_ZZ1 +_FO 1_MC1 ,_, to_II which_DDQ we_PPIS2 can_VM ascribe_VVI arbitrary_JJ values_NN2 ._. 
Here_RL ,_, we_PPIS2 shall_VM (_( a_ZZ1 )_) choose_VV0 as_II an_AT1 independent_JJ variable_NN1 ;_; (_( b_ZZ1 )_) prescribe_VV0 the_AT value_NN1 unity_NN1 for_IF &lsqb;_( formula_NN1 &rsqb;_) ;_; (_( c_ZZ1 )_) prescribe_VV0 the_AT value_NN1 zero_NN1 for_IF each_DD1 of_IO the_AT first_MD n_ZZ1 -_- 1_MC1 functions_NN2 in_II F_ZZ1 ,_, the_AT last_MD being_VBG written_VVN f_ZZ1 ._. 
Then_RT f_ZZ1 and_CC the_AT coordinates_NN2 &lsqb;_( formula_NN1 &rsqb;_) become_VV0 functions_NN2 of_IO ,_, and_CC we_PPIS2 note_VV0 that_CST when_CS f_ZZ1 vanishes_VVZ (_( i.e._REX all_DB the_AT functions_NN2 F_ZZ1 vanish_VV0 in_RP (_( 2_MC )_) )_) ,_, will_VM be_VBI an_AT1 eigenvalue_NN1 and_CC &lsqb;_( formula_NN1 &rsqb;_) the_AT corresponding_JJ eigenvector_NN1 ._. 
We_PPIS2 not_XX write_VV0 ,_, in_II partitioned_JJ form_NN1 ,_, &lsqb;_( formula_NN1 &rsqb;_) ,_, &lsqb;_( formula_NN1 &rsqb;_) ,_, and_CC ,_, since_CS the_AT pencil_NN1 in_II (_( 2_MC )_) is_VBZ symmetric_JJ ,_, &lsqb;_( formula_NN1 &rsqb;_) ,_, where_RRQ &lsqb;_( formula_NN1 &rsqb;_) is_VBZ a_AT1 square_JJ submatrix_NN1 of_IO K_ZZ1 of_IO order_NN1 n_ZZ1 =_FO 1_MC1 ._. 
Then_RT (_( 2_MC )_) becomes_VVZ &lsqb;_( formula_NN1 &rsqb;_) ,_, which_DDQ yields_VVZ ,_, progressively_RR ,_, &lsqb;_( formula_NN1 &rsqb;_) ,_, &lsqb;_( formula_NN1 &rsqb;_) ._. 
Since_CS the_AT submatrices_NN2 in_II (_( 5_MC )_) are_VBR all_DB functions_NN2 of_IO ,_, (_( 7_MC )_) and_CC (_( 8_MC )_) give_VV0 f_ZZ1 and_CC (_( or_CC x_ZZ1 )_) as_CSA functions_NN2 of_IO ._. 
Now_RT suppose_VV0 we_PPIS2 have_VH0 an_AT1 approximation_NN1 &lsqb;_( formula_NN1 &rsqb;_) to_II an_AT1 eigenvalue_NN1 ._. 
Since_CS f_ZZ1 is_VBZ a_AT1 function_NN1 of_IO which_DDQ vanishes_VVZ at_II the_AT eigenvalues_NN2 ,_, we_PPIS2 may_VM use_VVI it_PPH1 in_II the_AT Newton-Raphson_NP1 method_NN1 to_TO obtain_VVI an_AT1 improved_JJ approximation_NN1 &lsqb;_( formula_NN1 &rsqb;_) :_: &lsqb;_( formula_NN1 &rsqb;_) ,_, the_AT zero_NN1 suffixes_NN2 indicating_VVG that_CST the_AT functions_NN2 are_VBR calculated_VVN for_IF &lsqb;_( formula_NN1 &rsqb;_) ._. 
Equation_NN1 (_( 8_MC )_) is_VBZ not_XX suitable_JJ for_IF differentiation_NN1 ;_; instead_RR ,_, we_PPIS2 obtain_VV0 another_DD1 form_NN1 for_IF f_ZZ1 by_II premultiplying_VVG (_( 6_MC )_) by_II &lsqb;_( formula_NN1 &rsqb;_) ;_; we_PPIS2 obtain_VV0 &lsqb;_( formula_NN1 &rsqb;_) ,_, and_CC since_CS K_ZZ1 is_VBZ symmetric_JJ it_PPH1 follows_VVZ that_CST &lsqb;_( formula_NN1 &rsqb;_) ,_, which_DDQ ,_, with_IW K_ZZ1 given_VVN by_II (_( 5_MC )_) ,_, reduces_VVZ to_II &lsqb;_( formula_NN1 &rsqb;_) ._. 
In_II a_AT1 numerical_JJ calculation_NN1 ,_, we_PPIS2 use_VV0 (_( 11_MC )_) for_IF &lsqb;_( formula_NN1 &rsqb;_) and_CC either_RR (_( 8_MC )_) or_CC (_( 10_MC )_) for_IF f_ZZ1 ._. 
If_CS we_PPIS2 use_VV0 the_AT latter_DA ,_, we_PPIS2 must_VM note_VVI that_DD1 ,_, at_II &lsqb;_( formula_NN1 &rsqb;_) ,_, &lsqb;_( formula_NN1 &rsqb;_) ,_, when_CS Equation_NN1 (_( 9_MC )_) reduces_VVZ to_II &lsqb;_( formula_NN1 &rsqb;_) ,_, i.e._REX the_AT well-known_JJ Rayleigh_NP1 quotient_NN1 ._. 
In_II the_AT numerical_JJ evaluation_NN1 of_IO this_DD1 ,_, we_PPIS2 obtain_VV0 x_ZZ1 from_II (_( 7_MC )_) with_IW &lsqb;_( formula_NN1 &rsqb;_) ._. 
Having_VHG obtained_VVN &lsqb;_( formula_NN1 &rsqb;_) we_PPIS2 can_VM adopt_VVI it_PPH1 as_II a_AT1 new_JJ &lsqb;_( formula_NN1 &rsqb;_) and_CC repeat_VV0 the_AT cycle_NN1 of_IO approximation_NN1 ,_, which_DDQ is_VBZ known_VVN to_TO be_VBI quadratically_RR convergent_JJ :_: i.e._REX if_CS &lsqb;_( formula_NN1 &rsqb;_) is_VBZ the_AT true_JJ eigenvalue_NN1 ,_, then_RT for_IF small_JJ differences_NN2 &lsqb;_( formula_NN1 &rsqb;_) ._. 
Equation_NN1 (_( 9_MC )_) ,_, with_IW &lsqb;_( formula_NN1 &rsqb;_) given_VVN by_II (_( 8_MC )_) ,_, is_VBZ analytically_RR precisely_RR equivalent_JJ to_II (_( 12_MC )_) ._. 
In_II numerical_JJ applications_NN2 ,_, they_PPHS2 may_VM differ_VVI ,_, depending_II21 on_II22 the_AT accuracy_NN1 with_IW which_DDQ &lsqb;_( formula_NN1 &rsqb;_) and_CC are_VBR evaluated_VVN ._. 
Probably_RR (_( 12_MC )_) is_VBZ the_AT user_NN1 to_TO use_VVI ._. 
Example_NN1 1_MC1 In_II Equation_NN1 (_( 1_MC1 )_) ,_, let_VV0 &lsqb;_( formula_NN1 &rsqb;_) ._. 
Suppose_VV0 we_PPIS2 are_VBR given_VVN an_AT1 approximate_JJ eigenvalue_NN1 &lsqb;_( formula_NN1 &rsqb;_) ._. 
We_PPIS2 evaluate_VV0 &lsqb;_( formula_NN1 &rsqb;_) ,_, which_DDQ we_PPIS2 have_VH0 partitioned_VVN as_CSA in_II (_( 5_MC )_) ._. 
Then_RT is_VBZ given_VVN by_II (_( 7_MC )_) as_CSA &lsqb;_( formula_NN1 &rsqb;_) ,_, and_CC the_AT reader_NN1 may_VM check_VVI that_CST this_DD1 yields_VVZ &lsqb;_( formula_NN1 &rsqb;_) ._. 
We_PPIS2 give_VV0 this_DD1 to_II four_MC decimal_JJ places_NN2 only_RR :_: it_PPH1 is_VBZ not_XX necessary_JJ to_TO calculate_VVI to_II great_JJ accuracy_NN1 ,_, but_CCB once_RR found_VVN ,_, it_PPH1 must_VM be_VBI used_VVN consistently_RR ,_, and_CC calculations_NN2 made_VVD accurately_RR ._. 
We_PPIS2 deduce_VV0 that_CST for_IF &lsqb;_( formula_NN1 &rsqb;_) ,_, the_AT approximate_JJ eigenvector_NN1 is_VBZ &lsqb;_( formula_NN1 &rsqb;_) ,_, and_CC we_PPIS2 are_VBR able_JK to_TO calculate_VVI &lsqb;_( formula_NN1 &rsqb;_) ,_, &lsqb;_( formula_NN1 &rsqb;_) ,_, whence_RRQ &lsqb;_( formula_NN1 &rsqb;_) ._. 
If_CS we_PPIS2 round_VV0 this_DD1 off_RP to_II a_AT1 new_JJ &lsqb;_( formula_NN1 &rsqb;_) ,_, we_PPIS2 quickly_RR find_VV0 that_CST &lsqb;_( formula_NN1 &rsqb;_) also_RR ,_, i.e._REX 40_MC is_VBZ an_AT1 exact_JJ eigenvalue_NN1 ;_; also_RR the_AT corresponding_JJ eigenvector_NN1 becomes_VVZ &lsqb;_( formula_NN1 &rsqb;_) ._. 
If_CS ,_, instead_II21 of_II22 the_AT Rayleigh_NP1 quotient_NN1 ,_, we_PPIS2 use_VV0 (_( 9_MC )_) directly_RR ,_, we_PPIS2 require_VV0 from_II (_( 8_MC )_) &lsqb;_( formula_NN1 &rsqb;_) ._. 
Hence_RR (_( see_VV0 also_RR (_( 16_MC )_) )_) &lsqb;_( formula_NN1 &rsqb;_) ,_, in_II close_JJ agreement_NN1 with_IW (_( 16_MC )_) ._. 
Problem_NN1 formulated_VVN in_II31 terms_II32 of_II33 a_AT1 single_JJ matrix_NN1 If_CS we_PPIS2 premultiply_RR (_( 2.10.1.2_MC )_) by_II &lsqb;_( formula_NN1 &rsqb;_) and_CC write_VV0 &lsqb;_( formula_NN1 &rsqb;_) ,_, we_PPIS2 obtain_VV0 the_AT problem_NN1 in_II31 terms_II32 of_II33 the_AT characteristic_JJ matrix_NN1 of_IO A_ZZ1 :_: &lsqb;_( formula_NN1 &rsqb;_) ._. 
Note_VV0 that_CST here_RL A_ZZ1 is_VBZ ,_, in_RR21 general_RR22 ,_, not_XX symmetric_JJ ;_; however_RR ,_, it_PPH1 evidently_RR possesses_VVZ the_AT same_DA eigenvalues_NN2 as_CSA the_AT system_NN1 (_( 2.10.1.2_MC )_) ._. 
Let_VV0 us_PPIO2 now_RT abandon_VVI the_AT connection_NN1 with_IW a_AT1 mechanical_JJ system_NN1 ,_, and_CC treat_VV0 this_DD1 as_II a_AT1 problem_NN1 involving_VVG any_DD given_JJ real_JJ matrix_NN1 A_ZZ1 ,_, the_AT eigenvalues_NN2 and_CC vectors_NN2 for_IF which_DDQ have_VH0 to_TO be_VBI found_VVN ._. 
The_AT treatment_NN1 is_VBZ closely_RR similar_JJ to_II that_DD1 of_IO 2.10.1_MC ._. 
As_CSA before_RT ,_, we_PPIS2 choose_VV0 as_II an_AT1 independent_JJ variable_NN1 ,_, assign_VV0 the_AT value_NN1 unity_NN1 to_II &lsqb;_( formula_NN1 &rsqb;_) and_CC zeros_NN2 to_II the_AT first_MD n_ZZ1 -_- 1_MC1 elements_NN2 of_IO G_ZZ1 :_: &lsqb;_( formula_NN1 &rsqb;_) ;_; we_PPIS2 also_RR write_VV0 ,_, remembering_VVG that_CST A_ZZ1 is_VBZ not_XX symmetric_JJ ,_, &lsqb;_( formula_NN1 &rsqb;_) ._. 
Then_RT Equation_NN1 (_( 2_MC )_) gives_VVZ us_PPIO2 &lsqb;_( formula_NN1 &rsqb;_) ,_, from_II which_DDQ we_PPIS2 obtain_VV0 as_CSA before_II &lsqb;_( formula_NN1 &rsqb;_) ,_, &lsqb;_( formula_NN1 &rsqb;_) ._. 
We_PPIS2 now_RT look_VV0 for_IF the_AT left_JJ vector_NN1 &lsqb;_( formula_NN1 &rsqb;_) corresponding_VVG to_II x_ZZ1 as_II the_AT right_JJ vector_NN1 ;_; in_II 2.10.1_MC this_DD1 was_VBDZ &lsqb;_( formula_NN1 &rsqb;_) :_: here_RL it_PPH1 is_VBZ different_JJ ._. 
In_II31 place_II32 of_II33 (_( 2_MC )_) we_PPIS2 write_VV0 &lsqb;_( formula_NN1 &rsqb;_) ;_; by_II the_AT same_DA argument_NN1 as_CSA before_RT ,_, we_PPIS2 set_VV0 &lsqb;_( formula_NN1 &rsqb;_) ,_, &lsqb;_( formula_NN1 &rsqb;_) ,_, hence_RR from_II (_( 4_MC )_) ,_, (_( 9_MC )_) and_CC (_( 10_MC )_) &lsqb;_( formula_NN1 &rsqb;_) ,_, yielding_VVG &lsqb;_( formula_NN1 &rsqb;_) ,_, &lsqb;_( formula_NN1 &rsqb;_) ,_, and_CC on_II comparison_NN1 of_IO (_( 12_MC )_) with_IW (_( 7_MC )_) we_PPIS2 see_VV0 that_CST h_ZZ1 and_CC g_ZZ1 are_VBR identical_JJ ;_; this_DD1 is_VBZ not_XX surprising_JJ ,_, since_CS they_PPHS2 are_VBR evidently_RR similar_JJ functions_NN2 and_CC have_VH0 the_AT same_DA zeros_NN2 ,_, i.e._REX the_AT eigenvalues_NN2 ._. 
As_CSA before_RT ,_, we_PPIS2 now_RT employ_VV0 g_ZZ1 in_II a_AT1 Newton-Raphson_NP1 application_NN1 to_TO improve_VVI the_AT accuracy_NN1 of_IO any_DD given_JJ approximate_JJ eigenvalue_NN1 &lsqb;_( formula_NN1 &rsqb;_) ._. 
Premultiplication_NN1 of_IO (_( 5_MC )_) by_II &lsqb;_( formula_NN1 &rsqb;_) gives_VVZ &lsqb;_( formula_NN1 &rsqb;_) ,_, and_CC then_RT ,_, differentiating_JJ term_NN1 by_II term_NN1 ,_, we_PPIS2 have_VH0 &lsqb;_( formula_NN1 &rsqb;_) ,_, which_DDQ in_II31 view_II32 of_II33 (_( 4_MC )_) reduces_VVZ to_II &lsqb;_( formula_NN1 &rsqb;_) ._. 
Accordingly_RR ,_, as_CSA before_II &lsqb;_( formula_NN1 &rsqb;_) ._. 
This_DD1 is_VBZ the_AT parallel_NN1 to_II (_( 2.10.1.12_MC )_) ;_; as_CSA before_RT ,_, we_PPIS2 can_VM ,_, if_CS we_PPIS2 wish_VV0 ,_, use_VV0 g_ZZ1 as_CSA given_VVN by_II (_( 7_MC )_) for_IF alternative_JJ treatment_NN1 ,_, but_CCB we_PPIS2 shall_VM not_XX pursue_VVI this_DD1 here_RL ._. 
Example_NN1 1_MC1 In_II (_( 2_MC )_) ,_, let_VV0 &lsqb;_( formula_NN1 &rsqb;_) ._. 
This_DD1 matrix_NN1 is_VBZ ,_, in_II fact_NN1 ,_, obtained_VVN from_II our_APPGE previous_JJ example_NN1 by_II evaluating_VVG &lsqb;_( formula_NN1 &rsqb;_) ._. 
However_RR ,_, we_PPIS2 shall_VM treat_VVI a_AT1 different_JJ eigenvalue_NN1 :_: suppose_VV0 we_PPIS2 are_VBR told_VVN that_CST there_EX is_VBZ an_AT1 eigenvalue_NN1 in_II the_AT neighbourhood_NN1 of_IO &lsqb;_( formula_NN1 &rsqb;_) ._. 
We_PPIS2 evaluate_VV0 and_CC partition_NN1 &lsqb;_( formula_NN1 &rsqb;_) ._. 
Then_RT we_PPIS2 find_VV0 at_RR21 once_RR22 from_II (_( 6_MC )_) and_CC (_( 11_MC )_) that_CST &lsqb;_( formula_NN1 &rsqb;_) ,_, &lsqb;_( formula_NN1 &rsqb;_) ,_, and_CC hence_RR ,_, from_II (_( 15_MC )_) ,_, &lsqb;_( formula_NN1 &rsqb;_) ._. 
If_CS we_PPIS2 round_VV0 this_DD1 off_RP to_II a_AT1 new_JJ &lsqb;_( formula_NN1 &rsqb;_) ,_, and_CC repeat_VV0 the_AT calculation_NN1 we_PPIS2 leave_VV0 it_PPH1 to_II the_AT reader_NN1 to_TO check_VVI this_DD1 we_PPIS2 find_VV0 &lsqb;_( formula_NN1 &rsqb;_) ;_; finally_RR ,_, rounding_VVG this_DD1 off_RP to_II 20_MC and_CC repeating_VVG again_RT ,_, we_PPIS2 find_VV0 that_CST 20_MC is_VBZ an_AT1 exact_JJ eigenvalue_NN1 and_CC that_CST the_AT corresponding_JJ right_NN1 and_CC left_JJ eigenvectors_NN2 are_VBR ,_, respectively_RR ,_, &lsqb;_( formula_NN1 &rsqb;_) ,_, &lsqb;_( formula_NN1 &rsqb;_) ._. 
2.10.3_MC Factorisation_NN1 of_IO A_ZZ1 into_II two_MC symmetric_JJ matrices_NN2 In_II Equation_NN1 (_( 2.10.2.1_MC )_) we_PPIS2 derived_VVD A_ZZ1 as_II the_AT product_NN1 of_IO two_MC given_JJ symmetric_JJ matrices_NN2 ,_, &lsqb;_( formula_NN1 &rsqb;_) and_CC B_ZZ1 ;_; this_DD1 was_VBDZ done_VDN for_IF convenience_NN1 in_II the_AT examples_NN2 ._. 
For_IF completeness_NN1 ,_, we_PPIS2 interpolate_VV0 here_RL a_AT1 brief_JJ discussion_NN1 of_IO the_AT reverse_JJ procedure_NN1 :_: it_PPH1 was_VBDZ shown_VVN in_II 1.19_MC and_CC Theorem_NN1 V_ZZ1 of_IO 1.22_MC that_CST a_AT1 matrix_NN1 can_VM always_RR be_VBI factorised_VVN into_II two_MC symmetric_JJ matrices_NN2 ,_, but_CCB the_AT subject_NN1 was_VBDZ not_XX pursued_VVN there_RL ._. 
Here_RL we_PPIS2 obtain_VV0 the_AT most_RGT general_JJ solution_NN1 ._. 
We_PPIS2 first_MD continue_VV0 the_AT discussion_NN1 of_IO 1.19_MC ,_, and_CC treat_VV0 the_AT most_RGT common_JJ case_NN1 in_II which_DDQ A_ZZ1 has_VHZ distinct_JJ eigenvalues_NN2 ._. 
It_PPH1 was_VBDZ shown_VVN there_RL that_CST A_ZZ1 permutes_VVZ with_IW &lsqb;_( formula_NN1 &rsqb;_) ,_, which_DDQ is_VBZ therefore_RR diagonal_JJ ._. 
This_DD1 diagonal_JJ matrix_NN1 ,_, D_ZZ1 ,_, is_VBZ arbitrary_JJ ,_, since_II X_ZZ1 is_VBZ arbitrary_JJ to_II a_AT1 postmultiplying_JJ diagonal_JJ matrix_NN1 ._. 
Write&lsqb;formula&rsqb;_VV0 where_RRQ P_ZZ1 ,_, Q_ZZ1 ,_, R_ZZ1 ,_, ..._... are_VBR arbitrary_JJ ._. 
Now_RT if_CS (_( see_VV0 1.18_MC )_) we_PPIS2 write_VV0 &lsqb;_( formula_NN1 &rsqb;_) ,_, C_ZZ1 may_VM now_RT be_VBI expressed_VVN as_CSA &lsqb;_( formula_NN1 &rsqb;_) ,_, so_CS21 that_CS22 C_ZZ1 is_VBZ expressible_JJ as_II an_AT1 arbitrary_JJ linear_JJ sum_NN1 of_IO the_AT n_ZZ1 symmetric_JJ unit_NN1 rank_NN1 matrices_NN2 &lsqb;_( formula_NN1 &rsqb;_) ._. 
Next_MD consider_VV0 the_AT case_NN1 where_CS A_ZZ1 has_VHZ two_MC equal_JJ eigenvalues_NN2 ,_, but_CCB is_VBZ not_XX defective_JJ ._. 
Then_RT A_ZZ1 can_VM be_VBI written_VVN with_IW the_AT (_( 2_MC 2_MC )_) scalar_JJ submatrix_NN1 I_ZZ1 in_II the_AT leading_JJ diagonal_JJ position_NN1 ._. 
This_DD1 means_VVZ that_CST D_ZZ1 is_VBZ not_XX necessarily_RR diagonal_JJ ;_; corresponding_VVG to_II I_MC1 there_EX may_VM be_VBI a_AT1 (_( 2_MC 2_MC )_) block_NN1 which_DDQ is_VBZ arbitrary_JJ except_CS21 that_CS22 it_PPH1 must_VM be_VBI symmetric_JJ and_CC non-singular_NN1 ._. 
This_DD1 therefore_RR contains_VVZ three_MC arbitrary_JJ quantities_NN2 ,_, giving_VVG n_ZZ1 +_FO 1_MC1 in_II all_DB ._. 
Finally_RR ,_, suppose_VV0 A_ZZ1 has_VHZ two_MC equal_JJ eigenvalues_NN2 but_CCB is_VBZ defective_JJ ._. 
As_CSA previously_RR ,_, we_PPIS2 note_VV0 that_CST &lsqb;_( formula_NN1 &rsqb;_) since_CS &lsqb;_( formula_NN1 &rsqb;_) is_VBZ symmetric_JJ ._. 
Accordingly_RR &lsqb;_( formula_NN1 &rsqb;_) with_IW K_ZZ1 as_CSA defined_VVN in_II Theorem_NN1 V_ZZ1 of_IO 1.22_MC ._. 
Thus_RR &lsqb;_( formula_NN1 &rsqb;_) permutes_VVZ with_IW &lsqb;_( formula_NN1 &rsqb;_) ._. 
This_DD1 implies_VVZ that_CST &lsqb;_( formula_NN1 &rsqb;_) is_VBZ diagonal_JJ except_II21 for_II22 the_AT leading_JJ (_( 2_MC 2_MC )_) submatrix_VV0 ._. 
Now_RT the_AT following_JJ submatrices_NN2 permute_VV0 (_( see_VV0 also_RR 1.21_MC )_) &lsqb;_( formula_NN1 &rsqb;_) ,_, &lsqb;_( formula_NN1 &rsqb;_) ,_, where_CS a_AT1 ,_, b_ZZ1 are_VBR arbitrary_JJ ._. 
The_AT second_MD of_IO these_DD2 is_VBZ thus_RR the_AT leading_JJ submatrix_NN1 of_IO &lsqb;_( formula_NN1 &rsqb;_) ,_, whence_RRQ in_II &lsqb;_( formula_NN1 &rsqb;_) it_PPH1 is_VBZ &lsqb;_( formula_NN1 &rsqb;_) ,_, which_DDQ is_VBZ symmetric_JJ and_CC contains_VVZ just_RR two_MC arbitrary_JJ quantities_NN2 ;_; there_EX are_VBR thus_RR just_RR n_ZZ1 arbitrary_JJ quantities_NN2 in_II all_DB ._. 
Extension_NN1 to_II more_RGR complicated_JJ cases_NN2 is_VBZ not_XX difficult_JJ ._. 
However_RR ,_, these_DD2 studies_NN2 employ_VV0 the_AT modal_NN1 and_CC spectral_JJ matrices_NN2 ,_, and_CC in_II practice_NN1 these_DD2 may_VM not_XX be_VBI known_VVN ._. 
We_PPIS2 can_VM ,_, however_RR ,_, find_VVI directly_RR the_AT general_JJ solution_NN1 of_IO &lsqb;_( formula_NN1 &rsqb;_) ._. 
Write_VV0 &lsqb;_( formula_NN1 &rsqb;_) ,_, so_CS21 that_CS22 C_ZZ1 contains_VVZ n_ZZ1 (_( n_ZZ1 +_FO 1_MC1 )_) /2_MF unknown_JJ elements_NN2 ,_, n_ZZ1 in_II the_AT diagonal_JJ ,_, viz._REX p_ZZ1 ,_, q_ZZ1 ,_, r_ZZ1 ,_, ..._... ,_, and_CC n_ZZ1 (_( n_ZZ1 -_- 1_MC1 )_) /2_MF elsewhere_RL ,_, viz._REX a_AT1 ,_, b_ZZ1 ,_, c_ZZ1 ...._... 
Now_RT the_AT conditions_NN2 that_CST B_ZZ1 is_VBZ symmetric_JJ provide_VV0 just_RR n_ZZ1 (_( n_ZZ1 -1_MC )_) /2_MF linear_JJ algebraic_JJ equations_NN2 connecting_VVG the_AT unknowns_NN2 ,_, so_CS21 that_CS22 we_PPIS2 can_VM determine_VVI a_AT1 ,_, b_ZZ1 ,_, c_ZZ1 ,_, ..._... in_II31 terms_II32 of_II33 p_ZZ1 ,_, q_ZZ1 ,_, r_ZZ1 ..._... which_DDQ are_VBR arbitrary_JJ ._. 
The_AT procedure_NN1 is_VBZ best_RRT illustrated_VVN by_II31 means_II32 of_II33 a_AT1 numerical_JJ example_NN1 :_: we_PPIS2 choose_VV0 A_ZZ1 to_TO be_VBI the_AT matrix_NN1 of_IO (_( 2.10.2.16_MC )_) ._. 
Then_RT &lsqb;_( formula_NN1 &rsqb;_) ._. 
Then_RT the_AT condition_NN1 that_CST &lsqb;_( formula_NN1 &rsqb;_) ,_, for_REX21 example_REX22 ,_, leads_VVZ to_II &lsqb;_( formula_NN1 &rsqb;_) ,_, and_CC there_EX are_VBR five_MC other_JJ such_DA equations_NN2 ._. 
We_PPIS2 set_VV0 them_PPHO2 out_RP thus_RR :_: &lsqb;_( formula_NN1 &rsqb;_) These_DD2 are_VBR then_RT solved_VVN ,_, e.g._REX by_II direct_JJ operation_NN1 on_II rows_NN2 ,_, when_CS a_AT1 ,_, b_ZZ1 ,_, c_ZZ1 ,_, ..._... emerge_VV0 as_CSA quantities_NN2 linear_JJ in_II p_ZZ1 ,_, q_ZZ1 ,_, r_ZZ1 ,_, s_ZZ1 ._. 
To_TO set_VVI out_RP the_AT result_NN1 compactly_RR ,_, we_PPIS2 write_VV0 &lsqb;_( formula_NN1 &rsqb;_) ,_, where_RRQ &lsqb;_( formula_NN1 &rsqb;_) ..._... have_VH0 zero_MC diagonals_NN2 except_II21 for_II22 a_AT1 unit_NN1 in_II the_AT first_MD ,_, second_NNT1 ,_, ..._... place_VV0 respectively_RR ._. 
In_II this_DD1 example_NN1 ,_, we_PPIS2 can_VM write_VVI &lsqb;_( formula_NN1 &rsqb;_) ,_, where_RRQ to_TO keep_VVI the_AT elements_NN2 numerically_RR simple_JJ ,_, we_PPIS2 have_VH0 extracted_VVN the_AT factor_NN1 1/3_MF from_II &lsqb;_( formula_NN1 &rsqb;_) and_CC &lsqb;_( formula_NN1 &rsqb;_) ._. 
This_DD1 is_VBZ evidently_RR the_AT most_RGT general_JJ form_NN1 of_IO C._NP1 Postmultiplication_NN1 by_II A_ZZ1 yields_VVZ B_ZZ1 in_II the_AT form_NN1 &lsqb;_( formula_NN1 &rsqb;_) ._. 
When_CS p_ZZ1 ,_, q_ZZ1 ,_, r_ZZ1 ,_, s_ZZ1 are_VBR assigned_VVN numerically_RR ,_, C_ZZ1 and_CC B_ZZ1 are_VBR known_VVN and_CC then_RT the_AT factors_NN2 &lsqb;_( formula_NN1 &rsqb;_) and_CC B_ZZ1 of_IO A_ZZ1 are_VBR found_VVN ._. 
We_PPIS2 can_VM recover_VVI the_AT matrix_NN1 C_ZZ1 (_( and_CC also_RR B_ZZ1 )_) defined_VVD in_II (_( 2.10.1.13_MC )_) by_II assigning_VVG to_II p_ZZ1 ,_, q_ZZ1 ,_, r_ZZ1 ,_, s_ZZ1 the_AT values_NN2 1_MC1 ,_, 1.52_MC ,_, 3_MC ,_, 0.92_MC appearing_VVG in_II the_AT diagonal_JJ of_IO C._NP1 The_AT following_JJ characteristics_NN2 of_IO &lsqb;_( formula_NN1 &rsqb;_) ,_, &lsqb;_( formula_NN1 &rsqb;_) ,_, etc._RA are_VBR to_TO be_VBI noted_VVN ._. 
First_MD ,_, since_CS each_DD1 contains_VVZ zeros_NN2 in_II its_APPGE diagonal_JJ ,_, it_PPH1 can_VM not_XX be_VBI pos._NNU def_NN1 ._. 
Next_MD ,_, the_AT ranks_NN2 of_IO the_AT matrices_NN2 do_VD0 not_XX conform_VVI to_II any_DD simple_JJ pattern_NN1 :_: &lsqb;_( formula_NN1 &rsqb;_) is_VBZ of_IO rank_NN1 4_MC ,_, &lsqb;_( formula_NN1 &rsqb;_) and_CC &lsqb;_( formula_NN1 &rsqb;_) of_IO rank_NN1 3_MC and_CC &lsqb;_( formula_NN1 &rsqb;_) of_IO rank_NN1 2_MC ._. 
Finally_RR ,_, in_II31 relation_II32 to_II33 our_APPGE main_JJ problem_NN1 ,_, since_CS for_IF the_AT Rayleigh_NP1 quotient_NN1 (_( which_DDQ gives_VVZ eigenvalues_NN2 identical_JJ to_II those_DD2 of_IO A_ZZ1 )_) &lsqb;_( formula_NN1 &rsqb;_) ,_, and_CC since_CS also_RR p_ZZ1 ,_, q_ZZ1 ,_, r_ZZ1 ,_, s_ZZ1 are_VBR arbitrary_JJ ,_, it_PPH1 follows_VVZ that_CST &lsqb;_( formula_NN1 &rsqb;_) ,_, i_ZZ1 =_FO 1,2,3,4_MC ,_, though_CS this_DD1 result_NN1 is_VBZ not_XX particularly_RR helpful_JJ in_II practice._NNU 2.10.4_MC Another_DD1 method_NN1 for_IF improvement_NN1 of_IO eigenvalues_NN2 We_PPIS2 discuss_VV0 here_RL the_AT same_DA problem_NN1 as_CSA that_DD1 in_II 2.10.2_MC ,_, but_CCB vary_VV0 the_AT treatment_NN1 ._. 
The_AT eigenvalues_NN2 are_VBR determined_VVN by_II Equation_NN1 (_( 2.10.2.5_MC )_) with_IW g_ZZ1 =_FO 0_MC ,_, viz._REX &lsqb;_( formula_NN1 &rsqb;_) ,_, where_RRQ is_VBZ the_AT submatrix_NN1 of_IO A_ZZ1 corresponding_VVG to_II &lsqb;_( formula_NN1 &rsqb;_) ,_, and_CC &lsqb;_( formula_NN1 &rsqb;_) is_VBZ the_AT last_MD diagonal_JJ element_NN1 in_II A._NNU From_II this_DD1 we_PPIS2 find_VV0 &lsqb;_( formula_NN1 &rsqb;_) ,_, &lsqb;_( formula_NN1 &rsqb;_) ,_, and_CC elimination_NN1 of_IO leads_NN2 to_II the_AT equation_NN1 for_IF &lsqb;_( formula_NN1 &rsqb;_) ._. 
This_DD1 equation_NN1 is_VBZ not_XX easy_JJ to_TO solve_VVI numerically_RR as_CSA it_PPH1 stands_VVZ ;_; however_RR ,_, if_CS we_PPIS2 are_VBR given_VVN an_AT1 approximation_NN1 &lsqb;_( formula_NN1 &rsqb;_) to_II an_AT1 eigenvalue_NN1 ,_, we_PPIS2 write_VV0 &lsqb;_( formula_NN1 &rsqb;_) for_IF the_AT exact_JJ eigenvalue_NN1 ,_, which_DDQ therefore_RR satisfies_VVZ (_( 4_MC )_) ,_, so_CS21 that_CS22 we_PPIS2 obtain_VV0 &lsqb;_( formula_NN1 &rsqb;_) ,_, which_DDQ is_VBZ now_RT an_AT1 equation_NN1 for_IF Write_VV0 &lsqb;_( formula_NN1 &rsqb;_) then_RT &lsqb;_( formula_NN1 &rsqb;_) ._. 
Thus_RR &lsqb;_( formula_NN1 &rsqb;_) ,_, provided_VVN is_VBZ sufficiently_RR small_JJ ._. 
Insertion_NN1 in_II (_( 5_MC )_) leads_VVZ to_II &lsqb;_( formula_NN1 &rsqb;_) ,_, or_CC ,_, if_CS &lsqb;_( formula_NN1 &rsqb;_) ,_, &lsqb;_( formula_NN1 &rsqb;_) ,_, &lsqb;_( formula_NN1 &rsqb;_) ,_, etc._RA ,_, &lsqb;_( formula_NN1 &rsqb;_) ._. 
Finally_RR ,_, if_CS &lsqb;_( formula_NN1 &rsqb;_) is_VBZ written_VVN &lsqb;_( formula_NN1 &rsqb;_) ,_, we_PPIS2 obtain_VV0 the_AT scalar_JJ equation_NN1 &lsqb;_( formula_NN1 &rsqb;_) ._. 
Provided_CS the_AT series_NN converges_VVZ ,_, this_DD1 may_VM be_VBI solved_VVN for_IF in_II any_DD suitable_JJ way_NN1 ;_; often_RR a_AT1 regression_NN1 procedure_NN1 ,_, beginning_VVG with_IW =_FO 0_MC on_II the_AT right-hand_JJ side_NN1 to_TO give_VVI a_AT1 new_JJ approximation_NN1 on_II the_AT left_JJ ,_, will_VM solve_VVI the_AT equation_NN1 in_II two_MC or_CC three_MC steps_NN2 ._. 
Comparison_NN1 of_IO (_( 8_MC )_) with_IW (_( 3_MC )_) shows_VVZ that_CST ,_, when_RRQ has_VHZ been_VBN found_VVN from_II (_( 9_MC )_) &lsqb;_( formula_NN1 &rsqb;_) ,_, so_CS21 that_CS22 both_RR eigenvalue_VV0 and_CC vector_NN1 are_VBR found_VVN ._. 
Example_NN1 1_MC1 Once_RR21 again_RR22 ,_, we_PPIS2 use_VV0 the_AT matrix_NN1 specified_VVN in_II Equation_NN1 (_( 2.10.2.16_MC )_) and_CC we_PPIS2 are_VBR given_CS21 that_CS22 it_PPH1 has_VHZ an_AT1 eigenvalue_NN1 near_RL &lsqb;_( formula_NN1 &rsqb;_) ._. 
Then_RT ,_, in_II (_( 6_MC )_) ,_, with_IW n=4_FO ,_, &lsqb;_( formula_NN1 &rsqb;_) ._. 
Also_RR ,_, &lsqb;_( formula_NN1 &rsqb;_) ;_; &lsqb;_( formula_NN1 &rsqb;_) ;_; &lsqb;_( formula_NN1 &rsqb;_) ;_; ._. 
Hence_RR ,_, using_VVG (_( 7_MC )_) ,_, we_PPIS2 obtain_VV0 &lsqb;_( formula_NN1 &rsqb;_) ,_, &lsqb;_( formula_NN1 &rsqb;_) ,_, &lsqb;_( formula_NN1 &rsqb;_) ,_, &lsqb;_( formula_NN1 &rsqb;_) ,_, &lsqb;_( formula_NN1 &rsqb;_) ,_, and_CC it_PPH1 is_VBZ in_II fact_NN1 unnecessary_JJ to_TO proceed_VVI further_RRR ._. 
We_PPIS2 now_RT evaluate_VV0 the_AT scalars_NN2 &lsqb;_( formula_NN1 &rsqb;_) ;_; they_PPHS2 become_VV0 &lsqb;_( formula_NN1 &rsqb;_) ,_, &lsqb;_( formula_NN1 &rsqb;_) ,_, &lsqb;_( formula_NN1 &rsqb;_) ,_, &lsqb;_( formula_NN1 &rsqb;_) ,_, &lsqb;_( formula_NN1 &rsqb;_) ._. 
Hence_RR Equation_NN1 (_( 9_MC )_) ,_, for_IF ,_, becomes_VVZ &lsqb;_( formula_NN1 &rsqb;_) ,_, and_CC if_CS we_PPIS2 collect_VV0 the_AT linear_JJ terms_NN2 and_CC rationalise_VV0 :_: &lsqb;_( formula_NN1 &rsqb;_) and_CC solution_NN1 of_IO this_DD1 ,_, by_II regression_NN1 or_CC otherwise_RR ,_, gives_VVZ &lsqb;_( formula_NN1 &rsqb;_) ;_; &lsqb;_( formula_NN1 &rsqb;_) ._. 
Also_RR ,_, using_VVG (_( 10_MC )_) with_IW &lsqb;_( formula_NN1 &rsqb;_) ,_, we_PPIS2 find_VV0 &lsqb;_( formula_NN1 &rsqb;_) or_CC &lsqb;_( formula_NN1 &rsqb;_) ._. 
