MIT18_06SCF11_Ses2.8Eigenvalues and Eigenvectors.pdf

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1、EigenvaluesandeigenvectorsThesubjectofeigenvaluesandeigenvectorswilltakeupmostoftherestofthecourse.Wewillagainbeworkingwithsquarematrices.Eigenvaluesarespecialnumbersassociatedwithamatrixandeigenvectorsarespecialvectors.EigenvectorsandeigenvaluesAmatrixAactsonvectorsxlikeafunctiondoes

2、,withinputxandoutputAx.EigenvectorsarevectorsforwhichAxisparalleltox.Inotherwords:Ax=λx.Inthisequation,xisaneigenvectorofAandλisaneigenvalueofA.Eigenvalue0Iftheeigenvalueλequals0thenAx=0x=0.Vectorswitheigenvalue0makeupthenullspaceofA;ifAissingular,thenλ=0isaneigenvalueofA.ExamplesSupp

3、osePisthematrixofaprojectionontoaplane.ForanyxintheplanePx=x,soxisaneigenvectorwitheigenvalue1.AvectorxperpendiculartotheplanehasPx=0,sothisisaneigenvectorwitheigenvalueλ=0.TheeigenvectorsofPspanthewholespace(butthisisnottrueforeverymatrix).����011ThematrixB=hasaneigenvectorx=witheige

4、nvalue1101��1andanothereigenvectorx=witheigenvalue−1.Theseeigenvectors−1spanthespace.TheyareperpendicularbecauseB=BT(aswewillprove).det(A−λI)=0Annbynmatrixwillhaveneigenvalues,andtheirsumwillbethesumofthediagonalentriesofthematrix:a11+a22+···+ann.Thissumisthetraceofthematrix.Foratwoby

5、twomatrix,ifweknowoneeigenvaluewecanusethisfacttofindthesecond.CanwesolveAx=λxfortheeigenvaluesandeigenvectorsofA?Bothλandxareunknown;weneedtobeclevertosolvethisproblem:Ax=λx(A−λI)x=0Inorderforλtobeaneigenvector,A−λImustbesingular.Inotherwords,det(A−λI)=0.Wecansolvethischaracteristiceq

6、uationforλtogetnsolutions.1Ifwe’relucky,thesolutionsaredistinct.Ifnot,wehaveoneormorerepeatedeigenvalues.Oncewe’vefoundaneigenvalueλ,wecanuseeliminationtofindthenullspaceofA−λI.ThevectorsinthatnullspaceareeigenvectorsofAwitheigenvalueλ.Calculatingeigenvaluesandeigenvectors��31LetA=.The

7、n:13����3−λ1��det(A−λI)=�13−λ�=(3−λ)2−1=λ2−6λ+8.Notethatthecoefficient6isthetrace(sumofdiagonalentries)and8isthedeterminantofA.Ingeneral,theeigenvaluesofatwobytwomatrixarethesolutionsto:λ2−trace(A)·λ+detA=0.Justasthetraceisthesumoftheeigenvaluesofamatrix,theproductoftheeigenvaluesofany

8、matri