5) Suppose b=[00-23], c=[ 1-111], c=[202-1] and c3=[-3-1-33]T. Find x that minimizes || Ax - b||2...

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5) Suppose b=[00-23], c=[ 1-111], c=[202-1] and c3=[-3-1-33]T. Find x that minimizes || Ax - b||2 for the following two cases: a). b). A = A = [C C] A = A3 = [C C C3] 6) For a diagonalizable matrix A with order n, find constants a, a2, , an and rank-1 matrices A, A2,, An such that A = - aj Aj. 7) For a square matrix A, show that A is not diagonalizable if and only if AT is not diagonalizable. 8) Suppose A is real and symmetric and (X) = |A XD|, where D is diagonal and positive definite. Show that all roots of f are real. 9) Let A be a rank-1 matrix and I the identity matrix. Show that |A + I| = tr(A) + 1, where tr(A) is the trace of A. 10) Apply the singular value decomposition to show that A Rmxn with rank r can be written as A BC where B = R, C = Rrn, and both B and C have rank r. = 11) Suppose ai is the (i, j)th entry of A Rmxn. Find the sum of squares of singular values of A. 12) Suppose the matrix A is real and symmetric, and the first entry of every eigenvector of A is nonzero. Show that every algebraic multiplicity of the eigenvalue of A is 1. 5) Suppose b=[00-23], c=[ 1-111], c=[202-1] and c3=[-3-1-33]T. Find x that minimizes || Ax - b||2 for the following two cases: a). b). A = A = [C C] A = A3 = [C C C3] 6) For a diagonalizable matrix A with order n, find constants a, a2, , an and rank-1 matrices A, A2,, An such that A = - aj Aj. 7) For a square matrix A, show that A is not diagonalizable if and only if AT is not diagonalizable. 8) Suppose A is real and symmetric and (X) = |A XD|, where D is diagonal and positive definite. Show that all roots of f are real. 9) Let A be a rank-1 matrix and I the identity matrix. Show that |A + I| = tr(A) + 1, where tr(A) is the trace of A. 10) Apply the singular value decomposition to show that A Rmxn with rank r can be written as A BC where B = R, C = Rrn, and both B and C have rank r. = 11) Suppose ai is the (i, j)th entry of A Rmxn. Find the sum of squares of singular values of A. 12) Suppose the matrix A is real and symmetric, and the first entry of every eigenvector of A is nonzero. Show that every algebraic multiplicity of the eigenvalue of A is 1.

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5a To minimize Axb you can use the formula x AT A1 AT b where AT represents the transpose of matrix ... View the full answer