-1 3.) Given the matrix A 0 -2 2 1 (a) Find the singular values o; of A. [Hint: first show that the characteristic polynomial of (AT A) is X³ – 13 X² +36 A.] (b) Find the rank 1 matrix A' that is closest to A (with respect to the Frobenius norm). Note: you do NOT necd to find the whole SVD to do this. Find only those parts you need for A'. (c) Find both the Frobenius norm ||A||F and the 2-norm ||A||2 of A. Note: the matrix 2-norm is subordinate to the vector 2-norm.

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2.2.3 Compute the principal square root of each of the following positive semidefinite matrices. (b) 2 3 3 (d) 2 1 1 2 1 mi 1 1 (f) 2 2 0. 2 2 0 0 1 V3 0 0 V3 1,

7. Let A be an n x n Hermitian matrix. (a) Show that (Au, v) = (u, Av) for any n-dim vectors u and v. Here (, ·) denotes the inner product. (Hint: rewrite the inner product as matrix multiplication.) (b) Let x be an eigenvector associated to the eigenvalue A. Based on part (a), show that A(x, x) = X(x, x). (c) As in part (b), show that = X; that is, the eigenvalue A is real. 1