3.2 Diagonalization - Diagonalization Definition B A matrix...

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Diagonalization Definition. A matrix B is said similar to a matrix A if there is a nonsingular matrix P such that B = P 1 ∙ A∙P . Example 1. Let A = [ 1 1 2 4 ] and P = [ 1 1 1 2 ] . Then | P | = 1, A 11 = 2 , A 12 =− 1 , A 21 =− 1 , A 22 = 1 P 1 = [ 2 1 1 1 ] and B = P 1 ∙ A∙P = [ 2 1 1 1 ][ 1 1 2 4 ][ 1 1 1 2 ] = [ 4 2 3 3 ][ 1 1 1 2 ] = [ 2 0 0 3 ] . Thus B is similar to A . Similar matrices have the following elementary properties: 1. A is similar to A . 2. If B is similar to A , then A is similar to B . 3. If A is similar to B , then B is similar to C, then A is similar to C .
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Definition. We shall say that a matrix A is diagonalizable if it is similar to a diagonal matrix. Theorem 1. Similar matrices have the same eigenvalues . Proof. Let A and B be similar. Then B = P 1 ∙ A∙P and P 1 ¿ P det ( λI n A ) det ( ¿ )= det ( P ) ¿ ( P 1 ( λ I n A ) P ) = ¿ det ¿ ( λ I n P 1 ∙ A∙ P ) = ¿ det ( P 1 ∙λ I n ∙P P 1 ∙ A∙ P ) = det ¿ f B ( λ ) = det ( λ I n B ) = det ¿ Theorem 2. An n×n matrix A = [ a ij ] is diagonalizable if and only if
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