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Question 1 Consider the Hermitian matrix A. 2 - i 0 A - i 2 0 0 0 2 (a) Find a unitary matrix U such that A - UDU*, where D is a diagonal matrix. (b) Explain why A is positive definite, and compute the unique positive square root of A. You may leave your answer as a product of matrices. (c) Compute the Frobenius norm of A, i.e., A F - trace ( A * A). (d) Compute the operator norm of A, i.e., Il A| - sup{||xx|| : Il x < 1}. Question 2 Let N be an n X n complex matrix. (a) Show that if N is normal, then Il Nx Il Il N * x Il for all x € Cn. (b) Show that N is normal if and only if N + al is normal for all a € C. (c) Use parts (a) and (b) to show that if N is a normal and U € Cn is an eigenvector of N with eigenvalue 1, then U is an eigenvector of N* with eigenvalue T. Question 3 Let C([-1,1]) be the vector space of continuous functions f: [-1,1] C with inner-product 1 <f,g) =="" i="" f(t)g="" (t)="" dt.<br="">-1 Suppose that W - span {t, t2, A}. (a) Use the Gram-Schmidt algorithm to find an orthonormal basis for W. (b) Find the function g € W closest to the constant function f (x) I 1 for all x € [ - -1,1]. </f,g)>

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