HW14-solutions - huang(dh34953 HW14 gilbert(54160 This...

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huang (dh34953) – HW14 – gilbert – (54160) 1 This print-out should have 13 questions. Multiple-choice questions may continue on the next column or page – find all choices before answering. 001(part1of3)10.0points When A = U Σ V T is a Singular Value De- composition of the matrix A = bracketleftBigg 2 3 0 2 bracketrightBigg , determine V . 1. V = bracketleftbigg 2 - 1 1 2 bracketrightbigg 2. V = 1 5 bracketleftbigg 1 - 2 2 1 bracketrightbigg correct 3. V = 1 5 bracketleftbigg 2 - 1 1 2 bracketrightbigg 4. V = bracketleftbigg 1 - 2 2 1 bracketrightbigg Explanation: Since A T A = bracketleftbigg 2 0 3 2 bracketrightbigg bracketleftbigg 2 3 0 2 bracketrightbigg = bracketleftbigg 4 6 6 13 bracketrightbigg , the eigenvalues of A T A are the solutions of (4 - λ )(13 - λ ) - 36 = λ 2 - 17 λ + 16 = ( λ - 16)( λ - 1) = 0 , i.e. , λ 1 = 16 , λ 2 = 1. Eigenvectors x 1 and x 2 associated with λ 1 and λ 2 are x 1 = bracketleftbigg 1 2 bracketrightbigg , x 2 = bracketleftbigg - 2 1 bracketrightbigg ; these are orthogonal. The associated or- thonormal eigenvectors are thus v 1 = 1 5 bracketleftbigg 1 2 bracketrightbigg , v 2 = 1 5 bracketleftbigg - 2 1 bracketrightbigg . Consequently, V = [ v 1 v 2 ] = 1 5 bracketleftbigg 1 - 2 2 1 bracketrightbigg . 002(part2of3)10.0points Determine the singular values of A listed in descending order. 1. σ 1 = 16 , σ 2 = 1 2. σ 1 = - 1 , σ 2 = 16 3. σ 1 = - 2 , σ 2 = 4 4. σ 1 = 4 , σ 2 = 1 correct Explanation: The singular values of A are σ 1 = radicalbig λ 1 and σ 2 = radicalbig λ 1 . But by Part 1, λ 1 = 16 and λ 2 = 1. Thus in descending order, σ 1 = 4 , σ 2 = 1 . 003(part3of3)10.0points Determine the matrix U in the Singular Value Decomposition A = U Σ V T of A . 1. U = bracketleftbigg 1 - 2 2 1 bracketrightbigg 2. U = 1 5 bracketleftbigg 2 - 1 1 2 bracketrightbigg correct 3. U = 1 5 bracketleftbigg 1 - 2 2 1 bracketrightbigg 4. U = bracketleftbigg 2 - 1 1 2 bracketrightbigg Explanation: The matrix U is given by U = [ u 1 u 2 ] where u 1 = 1 σ 1 A v 1 = 1 4 5 bracketleftbigg 2 3 0 2 bracketrightbigg bracketleftbigg 1 2 bracketrightbigg = 1 5 bracketleftbigg 2 1 bracketrightbigg , and u 2 = 1 σ 2 A v 2 = 1 5 bracketleftbigg 2 3 0 2 bracketrightbigg bracketleftbigg - 2 1 bracketrightbigg = 1 5 bracketleftbigg - 1 2 bracketrightbigg .
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huang (dh34953) – HW14 – gilbert – (54160) 2 Consequently, U = 1 5 bracketleftbigg 2 - 1 1 2 bracketrightbigg , giving a Singular Value Decomposition 1 5 bracketleftbigg 2 - 1 1 2 bracketrightbigg bracketleftbigg 4 0 0 1 bracketrightbigg 1 5 bracketleftbigg 1 2 - 2 1 bracketrightbigg for the matrix A = bracketleftBigg 2 3 0 2 bracketrightBigg , as can easily be checked by direct calculation. 004 10.0points Determine the singular value σ 1 for the ma- trix A = 2 0 - 4 0 - 1 0 4 0 - 2 .
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