## Transcribed Text

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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