d Form an approximation to A by truncating the last q terms in the singular

# D form an approximation to a by truncating the last q

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(d) Form an approximation to A by truncating the last q terms in the singular value de- composition: A 0 = R - q X k =1 σ k u k v T k . Apply the new pseudo-inverse A 0† to yn and plot the result. Try a number of different values of q , and choose the one which “looks best” to turn in (indicate the value of q used). Calculate the mean-square reconstruction error. (e) Now form another approximate inverse using Tikhonov regularization. Try a number of different values for δ and choose the one which “looks best” to turn in (indicate the value of δ used). Calculate the mean-square reconstruction error. (f) Summarize your findings by comparing the MSE in parts (c), (d), and (e). Also include the error of doing nothing: k x - yn 0 k 2 2 where yn 0 is the appropriate piece of yn . 2 Last updated 14:22, November 7, 2019

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4. Let A = 1 . 01 0 . 99 0 . 99 0 . 98 (a) Find the eigenvalue decomposition of A . Recall that λ is an eigenvalue of A if for some u [1] , u [2] (entries of the corresponding eigenvector) we have (1 . 01 - λ ) u [1] + 0 . 99 u [2] = 0 . 99 u [1] + (0 . 98 - λ ) u [2] = 0 . Another way of saying this is that we want the values of λ such that A - λ I (where I is the 2 × 2 identity matrix) has a non-trivial null space — there is a nonzero vector u such that ( A - λ I ) u = 0. Yet another way of saying this is that we want the values of λ such that det( A - λ I ) = 0. Once you have found the two eigenvalues, you can solve the 2 × 2 systems of equations Au 1 = λ 1 u 1 and Au 2 = λ 2 u 2 for u 1 and u 2 . (b) If y = 1 1 T , determine the solution to Ax = y .
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