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rec5 - Sam Burden Linear Systems(EE 221 Recitation#5 1...

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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 , 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) Show that the set of solutions of (#) is S = { x = x 0 + z : z ∈ N ( A ) } .
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