Scientific Computing

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CS205 – Class 6 Reading: Heath 3.6 (p137-143), 4.7 (p202) Singular Value Decomposition (SVD) 1. The Singular Value Decomposition is an eigenvalue-like decomposition for square n m × matrices. It has the form T A U V = where U is an m m × orthogonal matrix, V is an n n × orthogonal matrix, and is an m n × diagonal matrix with positive diagonal entries that are called the singular values of A. The columns of U and V are the singular vectors . a. Introduced and rediscovered many times: Beltrami in 1873, Jordan in 1875, Sylvester in 1889, Autonne in 1913, Eckart and Young in 1936. b. Pearson introduced principle component analysis (PCA) in 1901. It uses SVD. c. Numerical work by Chan, Businger, Golub, Kahan, etc. 2. The singular value decomposition of 1 2 3 4 5 6 7 8 9 10 11 12 A = is given by .141 .825 .420 .351 25.5 0 0 .504 .574 .644 .344 .426 .298 .782 0 1.29 0 .761 .057 .646 .547 .028 .664 .509 0 0 0 .408 .816 .408 .750 .371 .542 .079 0 0 0 ⎤⎡ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥ ⎣ ⎦⎣ .
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  • Fall '07
  • Fedkiw
  • Singular value decomposition, singular values, AAT

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