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5.3 - 5.3 Diagonalization Recall A diagonalizable means...

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5.3 Diagonalization Recall: A diagonalizable means there is invertible P so that for a diagonal matrix D , P - 1 AP = D or A = PDP - 1 or AP = PD It’s easy to then compute powers of A : A k = ( PDP - 1 )( PDP - 1 ) ··· ( PDP - 1 ) = PD k P - 1 EX: Claim A = 1 3 3 - 3 - 5 - 3 3 3 1 , P = 1 - 1 - 1 - 1 1 0 1 0 1 P - 1 AP = D = 1 0 0 0 - 2 0 0 0 - 2 Verify this: (1) AP = 1 3 3 - 3 - 5 - 3 3 3 1 1 - 1 - 1 - 1 1 0 1 0 1 = 1 2 2 - 1 - 2 0 1 0 - 2 1

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PD = 1 - 1 - 1 - 1 1 0 1 0 1 1 0 0 0 - 2 0 0 0 - 2 = 1 2 2 - 1 - 2 0 1 0 - 2 or (2) Can ﬁnd P - 1 = 1 1 1 1 2 1 - 1 - 1 0 – verify this yourself directly and then that A = PDP - 1 . To ﬁnd A 10 , for instance, A 10 = P 1 0 0 0 ( - 2) 10 0 0 0 ( - 2) 10 P - 1 A 10 = 1 - 1 - 1 - 1 1 0 1 0 1 1
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5.3 - 5.3 Diagonalization Recall A diagonalizable means...

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