Question: Problem 7.2 (8 points). Let A E Rnxn be a symmetric matrix. We say that A is positive semidefinite if x Ax 2 0 for

Problem 7.2 (8 points). Let A E Rnxn be a symmetric matrix. We say that A is "positive semidefinite" if x Ax 2 0 for all x E R". (a) Let A = UDU be a decomposition guaranteed by the Spectral Theorem. Prove that A is positive semidefinite if and only if y Dy 2 0 for all y E Rn. (b) Use part (a) to prove that A is positive semidefinite if and only if all the eigenvalues of A are non-negative. (c) Let J be the n x n matrix with all entries equal to 1. Prove that J is positive semidefinite. (d) Let M ERmxn. Prove that A = MTM ERnxn is positive semidefinite

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