4 a Let A be a N N symmetric matrix Show that 2 trace A N X n 1 � n where the λ

# 4 a let a be a n n symmetric matrix show that 2 trace

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4. (a) Let A be a N × N symmetric matrix. Show that 2 trace( A ) = N X n =1 λ n , where the { λ n } are the eigenvalues of A . (b) Recall the definition of the Frobenius norm of an M × N matrix: k A k F = M X m =1 N X n =1 | A [ m, n ] | 2 ! 1 / 2 . Show that k A k 2 F = trace( A T A ) = R X r =1 σ 2 r , where R is the rank of A and the { σ r } are the singular values of A . (c) The operator norm (sometimes called the spectral norm ) of an M × N matrix is k A k = max x R N , k x k 2 =1 k Ax k 2 . 1 The coefficients are ordered in x to match the MATLAB convention for polynomials. 2 The trace of a matrix is the sum of the elements on the diagonal: trace( A ) = N n =1 A [ n, n ]. 2 Last updated 11:21, October 31, 2019

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(This matrix norm is so important, it doesn’t even require a designation in its notation — if somebody says “matrix norm” and doesn’t elaborate, this is what they mean.) Show that k A k = σ 1 , where σ 1 is the largest singular value of A . For which x does k Ax k 2 = k A k · k x k 2 ? (d) Prove that k A k ≤ k A k F . Give an example of an A with k A k = k A k F . 5. Suppose we have a signal f ( t ) on [0 , 1] which is “bandlimited” in that it only has N = 2 B +1 Fourier series coefficients which are non-zero: f ( t ) = B X k = - B α k e j 2 πkt , (5) for some set of expansion coefficents α = α - B .
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