ME 501A Seminar in Engineering Analysis Page 1

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1 Mtr Trsformtos usg Egevectors September 8, Mtr Trsformtos Usg Egevectors Lrry Cretto Mechcl Egeerg A Semr Egeerg Alyss September 8, Outle Revew lst lecture Trsformtos wth mtr of egevectors: = - A ermt d orthogol mtrces Qudrtc forms Numercl methods for egevlues d egevectors Revew Eges Bsc defto A : A= Det[A I] = gves th order equto for egevlues egevlues my ot be dstct solve[a I] = for compoets of ech of egevectors egevectors udetermed to wth multplctve costt egevectors my or my ot be lerly depedet Trsform A to Dgol If the egevectors of A re lerly depedet we c defe vertble mtr,, whose colums re egevectors of A: = [.. ] A = [A A A.. A ] A = [.. ] We ow show tht A = D where s dgol mtr of egevlues 4 Λ Mtr Product Usul mtr compoet s, row,colum Ths compoet otto s vectorrow Usul mtr multplcto formul pples Mtr Product Cotued Λ A A = [.. ] from prevous slde. We ow see tht A = ME A Semr Egeerg Alyss Pge

2 Mtr Trsformtos usg Egevectors September 8, ME A Semr Egeerg Alyss Pge A Trsformed We ssumed tht hs verse; we c pre-multply A = by ths verse to get Λ Λ A A 8 Trsform Emple for Lst clss emple: egevectors for mtr A, wth = d = A The mtr d ts verse re 9 Chec Iverse, Comupte A I I? Does A Trsform Emple Result Ths emple produces the epected result: - A s dgol mtr of egevlues regrdless of d A Aother - A Emple Lst clss emple hd A mtr wth egevlues = -, = -, = d egevectors show below A 4 Aother - A Emple II Usg mtrces - d A from the prevous slde we hve A 8 8 A

3 Mtr Trsformtos usg Egevectors September 8, Orthogol Mtrces A orthogol mtr hs mutully orthogol colums, Wrte mtr s [ ], = T = = Summto formul s equvlet to mtr multplcto of A T A = I Thus, A T = A - for orthogol mtrces Both rows d colums re orthogol More o Orthogol Mtrces A vector trsform wth orthogol mtr preserves the vector legth For y = A, wth A orthogol, y = y T y = A T A = T A T A Orthogol mtr: A T = A - so A T A = I So, y = T A T A = T I = T = Cocluso: whe y = A, wth A orthogol, y = 4 ermt/symmetrc Mtrces Symmetrc mtr: A = A T ermt mtr: A = A = A T = A A rel symmetrc mtr s ermt mtr lso clled self-dot For ermt mtr Egevlues re rel Egevectors form lerly depedet, orthogol bss set dmesos My hve comple egevectors for comple A Utry Mtr Alog of orthogol mtr for comple-vlues mtrces For utry mtr, U, U = U - I. e. for utry mtr we get the verse by tg the trspose d settg ll vlues of to Egevectors of ermt mtr, A Form orthogol mtr for rel-vlued A d utry mtr f A hs comple vlues ermt Egevectors ermt Emple Recll mtr whose colums re egevectors gvg = - A Requres to hve verse Ths s gurteed for ermt A Furthermore, sce colums re orthogol egevectors, - =, whch s the sme s T for rel A Emple of rel ermt mtr, A A Solve Det[A I] = for egevlues = 4.9, = -.894, d =. SolveA I = for ut egevectors d costruct mtr ME A Semr Egeerg Alyss Pge

4 Mtr Trsformtos usg Egevectors September 8, ermt Emple Cotued Show egevectors re orthoorml T, C show, = 9 ermt Emple Cocluded Sce colums of re orthogol, s orthogol mtr: - = T C verfy ths by tg verse C lso show tht - A = dg[,, ] Aother ermt Emple Aother ermt Emple II Fd such tht - A = for A = A A Det A I Det A I Now tht egevlues re ow fd egevectors from [A I] = A Get egevector compoets for ech of the egevlues For =, = - =,, Aother ermt Emple III Apply geerl equto to = New row = Row Soluto s =, + =, d Old Row =, for y vlue of For =, = - Aother ermt Emple IV Apply geerl equto to = Soluto s =, = b, d =, for y vlue of b 4 ME A Semr Egeerg Alyss Pge 4

5 Mtr Trsformtos usg Egevectors September 8, ME A Semr Egeerg Alyss Pge Aother ermt Emple V Apply geerl equto to = Soluto s = c, =, d = c, for y vlue of c Aother ermt Emple VI Geerl result c c Set = b = c = b Normlzed vectors Aother ermt Emple VII, Ier dot products of ule vectors re zero for orthogol set,, 8 Aother ermt Emple VIII Form mtr from dvdul egevectors - should equl T for orthogol egevector mtr Aother ermt Emple I Chec to see f - = T true f T = I 9 T T Aother ermt Emple A Λ Detls o et slde

6 Mtr Trsformtos usg Egevectors September 8, ME A Semr Egeerg Alyss Pge Aother ermt Emple I A Λ A Qudrtc Form Q = T A, wth symmetrc A Q = A wth ermt A Both the sme f A hs ll rel vlues A Qudrtc Form II A A 4 Qudrtc Form III Postve defte f A for y Postve semdefte f A > for y ermt mtr s postve defte semdefte f ll ts egevlues re postve or zero, A Smlrty Trsformtos Trsformtos mportt mtr opertos d umercl lyss I the smlrty trsform B = P - AP, B wll hve the sme egevlues s A B = => P - AP = Premultply by P to get PP - AP = P PP - AP = IAP = AP = P = P A egevectors re P Numercl Egevlue/vector Bsed o smlrty trsformtos ouseholder/gves trsformtos covert mtr to dgol form Use lbrry progrms such s LINPAK, Vsul Numercs IMSL lbrry or MATLAB for clcultos C get rge for egevlues by cluso theorems

7 Mtr Trsformtos usg Egevectors September 8, Gerschgor Icluso Theorem Provdes set of usully overlppg dss o the comple ple tht cot the egevlues, Apply to ech dgol elemet to get ds wth ceter d rdus comple ple by row sum Gerschgor Ds Emple All egevlues le dss costructed from equto o prevous chrt ermt mtr egevlues must le log rel s 8 ME A Semr Egeerg Alyss Pge

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