rec5 - Sam Burden Linear Systems (EE 221) Recitation #5 Sep...

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Sam Burden Linear Systems ( EE 221 ) Recitation #5 — Sep 24, 2010 1 Computational Complexity Definition An operation is either a division or a multiplication and subtraction. Exercise Show that Gaussian Elimination takes O ( n 3 ) operations to invert a matrix. Strassen’s Method leads to an O ( n log 2 7+ o (1) ) algorithm. Ax b Exercise In this problem we thoroughly investigate the linear system Ax = b (#) , A C m × n , rank A = r. Let the singular value decomposition of A be A = U Σ V * : Σ = ± Λ 0 0 0 ² , Λ = diag { σ 1 ,...,σ r } . Partition U and V as U = ( U 1 U 2 ) , V = ( V 1 V 2 ) , where U 1 C m × r and V 1 C n × r . Recall: columns of U 1 ,U 2 ,V 1 ,V 2 form orthonormal bases for R ( A ) , R ( A ) , N ( A ) , N ( A ). (a) First show by direct multiplication that A = U 1 Λ V * 1 (b) Show that (#) has a solution if and only if U * 2 b = 0. (c) Suppose the condition in (b) holds. Show that one solution of (#) is x 0 = V 1 Λ - 1 U * 1 b . (d)
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This note was uploaded on 04/11/2011 for the course EE 221A taught by Professor Clairetomlin during the Fall '10 term at University of California, Berkeley.

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rec5 - Sam Burden Linear Systems (EE 221) Recitation #5 Sep...

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