hw3A_98F - Homework #3_1998 Fall MEAM 501 Analytical...

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Unformatted text preview: Homework #3_1998 Fall MEAM 501 Analytical Methods in Mechanics and Mechanical Engineering (Due day : November 10) 1. Consider a rectangular matrix and a vector = ---- = 2 1 3 2 2 4 5 1 4 3 2 1 1 2 3 4 5 f A and (1) Find the rank of A , the null space of A , the range of A , and the 2-norm of A by using the singular value decomposition of A . A = 5 4 3 2 1 1 -2 3 -4 1-5 4 0 -2 2 f = 3 1 2 Rank = 3 U = 0.7917 0.5939 0.1429 0.0320 -0.2740 0.9612-0.6100 0.7565 0.2359 S = 7.9646 0 0 0 0 6.4382 0 0 0 0 5.4877 0 V = 0.8840 -0.1688 0.0904 0.4264 0.0832 0.9241 -0.0742 0.2090 -0.3000 0.3103 0.1491 0.6036 -0.7122 -0.1000 0.3359 0.1197 -0.7345 -0.4933 0.3000-0.0498 0.2847 0.2872 0.1549 0.9000 Null Space of the matrix A is spanned by the two vectors of the matrix V associated with the zero singular values, that is, the fourth and fifth column vectors : -- = -- = 9000 . 3000 . 1000 . 3000 . 1549 . 4933 . 7122 . 2090 . 4264 . 5 4 v v and Range of A is spanned by the three column vectors of the matrix U. 2 norm of the matrix A is the first singular value 9646 . 7 1 = s . (2) Compare the norm of the solution f A x \ * = with the one for . * ) ( f A x pinv = xstar = 0.2955 0.6250-0.4886 norm = 0.8466 x = 0.0832 0.4208 0.3210-0.1430 0.2236 norm = 0.5979 They are different, but both are solutions of the matrix equation. The solution by pinv is the solution with the least norm....
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hw3A_98F - Homework #3_1998 Fall MEAM 501 Analytical...

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