18 - Psuedoinverses

18 - Psuedoinverses - 1 Math 1b Practical March 7, 2011...

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1 Math 1b Practical — March 7, 2011 Singular value decomposition; psuedoinverses Recall that every symmetric matrix can be written as UDU > where D is nonnegative diagonal and U is orthogonal. Something similar can be done even if the matrix is not symmetric, and perhaps not even square. The topic of SVD deals with minimizing and approximating, and so it has many applications. Theorem 1. Let A be a m × n real matrix. Then there exist orthogonal matrices U and V of order m and n , and a nonnegative ‘diagonal matrix’ D ,sothat A = UDV > . Here D is m × n and ‘diagonal’ means that all entries are zero except possibly the ( i,i ) - entries where i min { m,n } . Proof: We prove the theorem only in the case that A is (square and) nonsingular here. Let V be any orthogonal matrix so that V > ( A > A ) V is diagonal and positive deF- nite (positive diagonal entries). Let D be a p.d. diagonal square root of this matrix, so V > ( A > A ) V = D 2 .Le t U = AV D 1 . That was quick, but we must check that A = > and U > U = I . And these are easy: > =( AV D 1 )( DV > )= AV V > = A, U > U D 1 V > A > )( AV D 1 D 1 ( V > ( A > A ) V ) D = D 1 D 2 D 1 = I. ut The diagonal entries of D are the positive square roots of the eigenvalues of A > A and are called the singular values of A . (Caution: this can be confusing because “singular” is part of the term “singular values” and is not being used as in “singular matrix”.) The SVD (singular value decomposition) A = > of an m × n matrix A displays orthonormal bases for R n and R m ,theco lumns e 1 ,..., e n of V and the columns f 1 f m of U A e i = d i f i (and A e i = 0 if i>m )andfor i =1 , 2 min { } ,where d i is the i -th diagonal entry of D .
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This note was uploaded on 11/10/2011 for the course MA 1B taught by Professor Aschbacher during the Winter '08 term at Caltech.

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18 - Psuedoinverses - 1 Math 1b Practical March 7, 2011...

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