cs357-slides-orthog

# cs357-slides-orthog - orthogonalization L Olson Department...

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orthogonalization L. Olson October 27, 2015 Department of Computer Science University of Illinois at Urbana-Champaign 1

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objectives Revisit SVD and Orthogonal Matrices Create orthogonal vectors Outline the Gram-Schmidt algorithm for orthogonalization 2
normal equations: conditioning The normal equations tend to worsen the condition of the matrix. Theorem cond ( A T A ) = ( cond ( A )) 2 1 A = np.random.rand(10,10) 2 print (np.linalg.cond(A)) 3 print (np.linalg.cond(A.T.dot(A))) 4 5 50.0972712517 6 2509.73658686 3

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other approaches QR factorization. For A R m × n , factor A = QR where Q is an m × m orthogonal matrix R is an m × n upper triangular matrix (since R is an m × n upper triangular matrix we can write R = " R 0 0 # where R is n × n upper triangular and 0 is the ( m - n ) × n matrix of zeros) SVD - singular value decomposition For A R m × n , factor A = USV T where U is an m × m orthogonal matrix V is an n × n orthogonal matrix S is an m × n diagonal matrix whose elements are the singular values.
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