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ECE 493 HW #3b – Version 1.13 March 18, 2011 Spring 2011 Univ. of Illinois Due Tu., Mar. 29 Prof. Allen Topic of this homework: LinearAlgebra: Solutions of non-symmetric matrices (Tall and Fat), Singular value decomposition (SVD) Deliverables: Show your work. Numerical results may not be sufficient, unless specifically requested. 1. Least-square solution of a tall matrix: (a) Find the least squares (LS) solution of the tall (over-specified) system of equations Ax = b 1 0 0 1 1 1 bracketleftbigg x 1 x 2 bracketrightbigg = 1 1 1 . Mentally note that the third equation (third row of A ) is very different than the first two, which would have the trivial solution x 1 = 1 and x 2 = 1. (b) Find the eigenvalues and eigenvectors of A T A for the A of part (a). (c) Repeat the entire process of (a) again, but this time the third equation is close to the first equation (let ǫ be a very small number): 1 0 0 1 1 ǫ bracketleftbigg x 1 x 2 bracketrightbigg = 1 1 1 . Summarize your conclusion about the impact of the third equation, as a function of
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Unformatted text preview: . Summarize your conclusion about the impact of the third equation, as a function of Ç« , on the LS solution. 2. Fat systems: Consider Ë† i , Ë† j , Ë† k as unit vectors in the x , y and z directions. (a) Consider the fat (under-specifed) system Ax = b b 1 2 Ç« 2-1-Ç« B x y z = b 1-1 B with Ç« a very small number. (a) Identify span( A ) and null( A )? (b) What can you say about the solution space? 3. Singular Value Decomposition (SVD) Let A = b 1 1 1-1 1 B and Â²nd U , Î£ and V such that A = U Î£ V T . Repeat the process on A T . Use the fact that U = eig ( AA T ) âˆˆ R m Ã— m and V = eig ( A T A ) âˆˆ R n Ã— n , where eig ( S ) are the eigenvectors of a symmetric matrix S . Version 1.13 March 18, 2011 Ëœ /493/Assignments/HW #3b â€“ Version 1.13 March 18, 2011...
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