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A6_soln - Math 235 1 1 Let A = 2 4 diagonalizes A 2 2 2 and...

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Math 235 Assignment 6 Solutions 1. Let A = 1 2 - 4 2 - 2 - 2 - 4 - 2 1 . Find an orthogonal matrix P that diagonalizes A and the corresponding diagonal matrix. Solution: The characteristic polynomails is C ( λ ) = det( A - λI ) = det 1 - λ 2 - 4 2 - 2 - λ - 2 - 4 - 2 1 - λ = det 1 - λ 2 - 4 2 - 2 - λ - 2 - 3 - λ 0 - 3 - λ = det 5 - λ 2 - 4 4 - 2 - λ - 2 0 0 - 3 - λ = - ( λ + 3)( λ 2 - 3 λ - 18) = - ( λ + 3) 2 ( λ - 6) . Hence, the eigenvalues of A are λ 1 = - 3 and λ 2 = 6. For λ 1 = - 3 we have A - λ 1 I = 4 2 - 4 2 1 - 2 - 4 - 2 4 1 1 / 2 - 1 0 0 0 0 0 0 . Thus, linearly independent eigenvectors corresonding to λ 1 are v 1 = - 1 2 0 , and v 2 = 1 0 1 . For λ 2 = 6 we have A - λ 2 I = - 5 2 - 4 2 - 8 - 2 - 4 - 2 - 5 1 0 1 0 1 1 / 2 0 0 0 . Thus, an eigenvector corresonding to λ 2 is - 2 - 1 2 . We first observe that eigenvectors for λ 1 are not orthogonal. So, applying the Gram-Schmidt procedure to { v 1 , v 2 } : Let w 1 = - 1 2 0 ; then w 2 = 1 0 1 - - 1 5 - 1 2 0 = 4
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