practiceprob9_soln

# practiceprob9_soln - Math 235 Assignment 9 Practice...

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Unformatted text preview: Math 235 Assignment 9 Practice Problems 1. Let A = 1- 1 1- 4- 4 1 . 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 = 18- 9- 9 18 . Hence, the eigenvalues of A T A are λ 1 = 27 and λ 2 = 9. Thus the maximum of k A~x k subject to k ~x k = 1 is √ 27 and the minimum is √ 9 = 3. 2. Find a singular value decomposition of each of the following matrices. a) A = 1- 2 2 2 2- 1 Solution: We have A T A = 9 0 0 9 . Hence, the eigenvalues are λ 1 = 9 and λ 2 = 9, so the singular values are σ 1 = 3 and σ 2 = 3. We take V = I . Then We let ~u 1 = 1 σ 1 A~v 1 = 1 3 1 2 2 ~u 2 = 1 σ 2 A~v 2 = 1 3 - 2 2- 1 Extend { ~u 1 ,~u 2 } to an orthonormal basis for R 3 using ~u 3 = ~u 1 × ~u 2 = 1 3 - 2- 1 2 . Thus, we take U = ~u 1 ~u 2 ~u 3 , Σ = 3 0 0 3 0 0 and V = I . We then have A = U Σ V T . 2 b) A = 4 4- 2 3 2 4 Solution: Let...
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practiceprob9_soln - Math 235 Assignment 9 Practice...

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