T 1 n x n x t n 1 p 1 n p n where i denotes the i th

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T 1 + · · · + n x n x T n = 1 P 1 + · · · + n P n where i denotes the i th eigenvalue of A with corresponding unit eigenvector x i and P i denotes the i th spectral projector. (a) Write the spectral factorization of the matrix A = 1 2 2 1 in the form 1 P 1 + 2 P 2 (b) Show that the matrices P 1 and P 2 are, in fact, orthogonal projections, i.e., show that P i = P T i = P 2 i . (c) Show that the orthogonal matrices P 1 and P 2 are orthogonal to each other, i.e., that P T 1 P 2 = P 1 P 2 = 0. 5
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5. (a) By verifying the four Penrose conditions, show that if A 2 IR m £ n has a singular value decomposition given by A = U ß V T (with all notation as given in the lecture notes), then A + = V ß + U T 2 IR n £ m where ß + = S ° 1 0 0 0 . (b) Repeat (a) for the “economy-sized” SVD A = U 1 SV T 1 , i.e., verify the four Penrose conditions for A + = V 1 S ° 1 U T 1 . 6
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6. Let A = 1 20 2 6 6 4 2 4 4 2 3 4 2 2 4 2 1 4 3 7 7 5 and suppose you compute (e.g., using Matlab ) an SVD U ß V T of A to produce the following matrices: U = 2 6 6 4 ° 0 . 5706 0 . 6119 ° 0 . 5000 0 . 2236 ° 0 . 5215 0 . 1674 0 . 8333 0 . 0745 ° 0 . 4724 ° 0 . 2771 ° 0 . 1667 ° 0 . 8199 ° 0 . 4233 ° 0 . 7217 ° 0 . 1667 0 . 5217 3 7 7 5 , ß = 2 6 6 4 0 . 5154 0 0 0 0 . 0970 0 0 0 0 0 0 0 3 7 7 5 , V = 2 4 ° 0 . 3857 ° 0 . 2263 0 . 8944 ° 0 . 5060 0 . 8625 0 . 0000 ° 0 . 7715 ° 0 . 4526 ° 0 . 4472 3 5 . Suppose that you also compute, from this data, that A + = 2 4 ° 1 0 1 2 6 2 ° 2 ° 6 ° 2 0 2 4 3 5 .
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  • Spring '08
  • VALDIMARSSON
  • Math, Multivariable Calculus, Singular value decomposition, Orthogonal matrix, Linear least squares, Prof. Alan J. Laub

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