Transcription of Lecture 34: Similar Matrices Math 2270
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Lecture 34: Similar Matrices Math 2270
Math2270-Lecture34 AS,whereAisthematrixoftheeigenvaluesofA, , ,9,10,12,131 SimilarMatricesTwomatricesAandBaresimila rifthereexistsaninvertiblematrixMsuchtha tB=M ,obviously,Aissimilartoitself,andifB=M AMthenA=MBAi- ,soifAissimilartoB, ,notethatifAM BM1andB=ii CPi9thenA=Ai (M CM2)M1(PvI2M1) C(M2M1).So,ifAissimilartoB,andBissimilar toC, Similar isreflexive,symmetric,andtransitive, regivenasetofobjects,andanequivalencerel ationontheseobjects,andyou ,andthey ,we ,we sright,bytheendoftheday,you , ,ifB=Ai ,thenM xisaneigenvectorofB=M ) wehaveMBA1 x=B(ii x)=So,A1 xisaneigenvectorforB,witheigenvalue).
Two similar matrices have the same eigenvalues, even though they will usually have different eigenvectors. Said more precisely, if B = Ai’AJ.I and x is an eigenvector of A, then M’x is an eigenvector of B = M’AM. The proof is quick. Suppose Ax )x. Then as A = A1BM’ we have
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