sec7-4 similarities

# We proceed as follows 1 find the eigenvalues of a ie

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is diagonal. We proceed as follows: 1. Find the eigenvalues of A , i.e., solve f ( λ ) = det ( λI n - A ) = 0. 2. For each eigenvalue λ , find a basis of the eigenspace E λ = ker ( λI n - A ). 3. A is diagonalizable iff the dimensions of the eigenspaces add up to n . In this case, we find an eigenbasis ~v 1 ,~v 2 , ...,~v n for A by combining the bases of the eigenspaces. Let S = h ~v 1 ~v 2 ... ~v n i , then the matrix S - 1 AS is a diagonal matrix. 3

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Example Diagonalize the matrix 1 1 1 0 0 0 0 0 0 Solution a. The eigenvalues are 0 and 1. b. E 0 = ker ( A ) = span ( - 1 1 0 , - 1 0 1 ) and E 1 = ker ( I 3 - A ) = span 1 0 0 c. If we let S = - 1 - 1 1 1 0 0 0 1 0 then D = S - 1 AS = 0 0 0 0 0 0 0 0 1
Alg 7.4.5 Powers of a diagonalizable ma- trix To compute the powers A t of a diagonalizable matrix A (where t is a positive integer), pro- ceed as follows: 1. Use Alg 7.4.4 to diagonalize A , i.e. find S such that S - 1 AS = D . 2. Since A = SDS - 1 , A t = SD t S - 1 . 3. To compute D t , raise the diagonal entries of D to the t th power. 4
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