Question

true or false and explain why

(a) If the eigenvalues of a real symmetric matrix Anxn are all positive, then 7" A7 > 0 for any i in R" (b) If a real square matrix is orthogonally diagonalizable, it must be symmetric. (c) If A is a real mx n matrix, then both APA and AA' are semi-positive definite. (d) SVD and orthogonal diagonalization coincide when the real matrix concerned is symmetric pos- itive definite. (e) If vectors and q in R" satisfy Ap = 2p and Aq = -77 for real symmetric matrix Anxn, then p=0.

Answer 1

a) true. Since the eigenvalues of real symmetric positive definite matrix are all positive.

b) true. This is a result , a matrix A is orthogonally diagonalizable and has real eigenvalues if and only if A is symmetric.

C) true.

d) true. For a positive definite symmetric matrixmatrixthe orthogonal diagonalization is a SVD of A.