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Unformatted text preview: Math 235 Assignment 9 Solutions 1. Let A = 1 2 3 1 2 . Find the maximum and minimum value of k A~x k for ~x ∈ R 2 subject to the constraint k ~x k = 1. Solution: We have A T A = 11 0 8 . Hence, the eigenvalues of A T A are λ 1 = 11 and λ 2 = 8. Thus the maximum of k A~x k subject to k ~x k = 1 is √ 11 and the minimum is √ 8. 2. Find a singular value decomposition of each of the following matrices. a) 1 1 1 1 Solution: Let A = 1 1 1 1 then A T A = 2 0 0 2 . Thus the only eigenvalue of A T A is λ 1 = 2 with multiplicity 2, and A is orthogonally diagonalized by V = 1 0 0 1 . The singular values of A are σ 1 = √ 2 and σ 2 = √ 2. Thus the matrix Σ is Σ = √ 2 √ 2 . Next compute ~u 1 = 1 σ 1 A~v 1 = 1 √ 2 1 1 1 1 1 = 1 1 ~u 2 = 1 σ 2 A~v 2 = 1 √ 2 1 1 1 1 1 = 1 1 Thus we have U = 1 / √ 2 1 / √ 2 1 / √ 2 1 / √ 2 . Then A = U Σ V T as required. 2 b) 1 4 2 2 2 4 Solution: Let A = 1 4 2 2 2 4 then A T A = 9 0 36 and the eigenvalues of...
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 Spring '08
 CELMIN
 Math, Linear Algebra, Singular value decomposition, singular values

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