In each part, you are given a matrix A Decide whether the...

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Problem Set 20 - Orthogonal Matrices, Symmetric Matrices and the Spectral Theorem 1. In each part, you are given a matrix A . Decide whether the matrix is orthogonally diagonalizable; if so, find an orthogonal matrix S and a diagonal matrix D such that A = SDS - 1 . (a) A = 4 - 1 - 2 0 9 0 - 10 - 2 5 (b) A = - 1 0 2
2 2 - 3 You may use the fact that the characteristic polynomial of A is (1 - λ ) 2 ( - 5 - λ ) .
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Gram-Schmidt to the basis ~v 1 = 1 - 1 0 ,~v 2 = 2 0 1 of E 1 : ~u 1 = 1 k ~v 1 k ~v 1 = 1 2 ~v 1 = 1 2 1 - 1 0 ~v 2 = ~v 2 - ( ~u 1 · ~v 2 ) ~u 1 = ~v 2 - 2 ~u 1 = 1 1 1 ~u 2 = 1 k ~v 2 k ~v 2 = 1 3 ~v 2 = 1 3 1 1 1 Since E - 5 is 1-dimensional, to get an orthonormal basis of E - 5 , we simply need a unit vector in E - 5 , so we can use

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