Solved: Set A = round(10 rand(10, 5)) and s = svd(A) (a) Use MATLAB to compute A 2, A F

Chapter 7, Problem 15

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Set A = round(10 rand(10, 5)) and s = svd(A) (a) Use MATLAB to compute A 2, A F, and cond2(A) and compare your results with s(1), norm(s), s(1)/s(5), respectively. (b) Set [ U, D, V ] = svd(A); D(5, 5) = 0; B = U D V The matrix B should be the closest matrix of rank 4 to A (where distance is measured in terms of the Frobenius norm). Compute A 2 and B 2. How do these values compare? Compute and compare the Frobenius norms of the two matrices. Compute also A B F and compare the result with s(5). Set r = norm(s(1 : 4)) and compare the result to B F. (c) Use MATLAB to construct a matrix C that is the closest matrix of rank 3 to A with respect to the Frobenius norm. Compute C 2 and C F. How do these values compare with the computed values for A 2 and A F, respectively? Set p = norm(s(1 : 3)) and q = norm(s(4 : 5)) Compute C F and A C F and compare your results with p and q, respectively.(i) AXA = A (ii) XAX = X(iii) (AX)T = AX (iv) (XA)T = XA(d) Compute and compare AX and U1(U1)T . Hadall computations been done in exact arithmetic,the two matrices would be equal. Why?Explain.