True or False. For all subquestions below, assume that A is an m times n real ma
ID: 3037511 • Letter: T
Question
True or False. For all subquestions below, assume that A is an m times n real matrix. (i) T F: Every eigenvalue of AA^T is nonnegative. (ii) T F: A A^T and A^T A have the same set of eigenvalues if not counting repeated zeros. (iii) T F: If A = U Sigma V^T is a singular value decomposition of A. Then columns of U are eigenvectors of A A^T, columns of V are eigenvectors of A^T A. (iv) T F: If x, y element R^n are eigenvectors of A and are linearly independent, then x and y are orthogonal. (v) T F: Let A and B l)e positive definite matrices (hence real symmetric). Then AB is also positive definite if and only if AB = BA.^2Explanation / Answer
(i). Let be an eigenvalue of AAT and let q be the corresponding eigenvector. Then (AAT)q = q. Now, on multiplying, to the left,by qT, we have qTAATq = qTq which implies that
=qTAATq/qTq = xTx/qTq, where x = ATq. Now, qTq > 0 and xTx 0. Therefore 0. This implies that all the eigenvalues of AAT are non-negative. The statement is true.
(ii) Let be an eigenvalue of AAT and let q be the corresponding eigenvector. Then (AAT)q = q. Now, on multiplying, to the left,by AT, we get AT(AAT)q = (ATA)·(ATq) = (ATq). hence is an eigenvalue of ATA with ATq as the corresponding eigenvector. Thus,ATA and AAT have the same set of non-zero eigenvalues. The statement is true.
(iii). If A = UVT is a singular value decomposition of A, then U and V are orthogonal matrices.The columns of U are eigenvectors of AAT and columns of V are eigenvectors of ATA.The statement is true.
(iv). The eigenvectors of a matrix A, even if linearly independent,need not be orthogonal. The statement is false.
(v).If A and B are positive definite matrices, then these are symmetric matrices with positive eigenvalues. Then AB also has positive eigenvalues, but (AB)T = BTAT = BA ( as A, and B are symmetric) = AB if and only if AB = BA. Thus, if A and B are positive definite matrices, then AB is also positive definite if and only if AB = BA. The statement is true.
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