MATA23 Lec18.pdf - March 18 � S I 15.2 maps and looked have we matrices represented Question Egn I Eigenvalues Eigenvectors characteristic In this class

MATA23 Lec18.pdf - March 18 � S I 15.2 maps and looked have...

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March 18 , 2021 § S . I 15.2 - characteristic Egn I Eigenvalues / Eigenvectors In this class we have looked at the relationship between 1in . maps and matrices . We know that any tin map can be represented as a matrix Wrt some basis . Question : Can we find a basis so that the matrix representing is diagonal ? Yes , sometimes . Diagonal matrices are best , because they are faster to work with ( add them , multiply , can check inner Libi lily faster ) Answering this question means we will look at the diagonal izatim problem . Note : say we have a 1in . map L : V V and we represent it Wrt the standard basis s of V and another basis B . This gives 2 different matrices that describe L as a matrix transformation : A - - [ Lfs and D= ChIp .
sometimes we can switch back and forth between A and B . Matrices that can do this are called similar Definition : Let A. BE IN nxn ( IR ) . A is said to be similar to B if there exists an invertible matrix PEIM nxn HR ) such that B = p - ' AP Note : * we can get back to A from B by going P ( B - - P " AP ) PB = Ppf ' AP Itn ( PB - - Ap ) p - i PBP " = APPT In A = PBP - I * The diagonal

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