diagonalize - 1 + d 2 + ··· d m elements in this list....

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How to diagonalize a matrix Let A be an n × n matrix. 1. Compute the characteristic polynomial f A ( x ) := det ( A - xA ) . This is a monic polynomial of degree n . 2. Find the roots λ 1 ...λ r m of f A ( X ), together with their multiplictiies m 1 ,...m r . There are at most n roots so r n . In fact m 1 + ··· m r = n , if you are willing to include complex roots if necessary. These roots are the eigenvalues of A . 3. For each i , find an ordered basis β i for the Eig λ i ( A ) = NS ( λ i I - A ), using Gauss elimination. Each β i will be a list of d i vectors, where d i is the dimension of Eig λ i ( A ). 4. Assemble all the bases you constructed above into single list β of vectors. There will be a total of d
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Unformatted text preview: 1 + d 2 + ··· d m elements in this list. 5. Theorem : The sequence β is automatically linearly independent. The matrix A is diagonalizable if and only if d 1 + ··· d r = n , and this is true if and only if d i = m i for all i . If this is the case, β is a basis for R n , and the matrix S whose columns are the vectors in β vectors satisfies AS = SD , with D diagonal. Note: Each d i ≥ 1, so if all the roots of f A ( X ) are distinct, then m = n , each d i = 1, ∑ d i = n , and A is automatically diagonalizable....
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This note was uploaded on 02/02/2011 for the course MAE 107 taught by Professor Tsao during the Spring '06 term at UCLA.

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