2. [Positive semi-definite matrices, 6pts] We have seen that the behavior of Hessians (i.e., second- derivative matrices) is important to study necessary and sufficient conditions for local minima. Thus, we review properties of symmetric matrices A E RX, i.e., A = A. Recall that such A can be diagonalized, i.e., there exists a matrix U E RXn which is orthogonal (i.e., UT = U-1) such that UTAU = A, where A := diag()1, 12, . .., An). Here, the di E R are the real eigenvalues of A, and the corresponding eigenvectors are the columns of U. (a) Show the equivalence of the following statements (in words: positive semi-definite matrices are characterized by non-negative eigenvalues): . d' Ad 0 for all d E Rn . d Ad 0 for all d E n . Xi 2 0 for all i = 1, ..., n. (b) Show the previous statement with all "2" replaced by "", and with the additional assump- tion that d * 0 (i.e., positive definite matrices are characterized by all positive eigenvalues). (c) Show that with the notation above, det(A) = IIt_1 dil (d) Show that with the notation above, the trace (i.e., the sum of the diagonal elements) is Tr(A) = En-1 di. 2 (e) Use the above arguments to argue that a symmetric matrix A E R2X2 is positive semi-definite if and only if det(A) 2 0 and Tr(A) 2 0. N (f) For which values of b E R is the matrix A = b 2 b positive semi-definite