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Unformatted text preview: CS205 – Class 5 Covered in class: 1, 3, 4, 5. Reading: Heath Chapter 4. Singular Value Decomposition (SVD) 1. The Singular Value Decomposition is an eigenvaluelike decomposition for rectangular 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 .504 .574 .644 .344 .426 .298 .782 1.29 .761 .057 .646 .547 .028 .664 .509 .408 .816 .408 .750 .371 .542 .079     . a. The singular values are 25.5, 1.29, and 0. The singular value of 0 indicates that the matrix is rank deficient. However, even a “small” singular value could indicate a “zero” and a rank deficient matrix. deficient....
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This note was uploaded on 01/05/2012 for the course CS 205A taught by Professor Fedkiw during the Fall '07 term at Stanford.
 Fall '07
 Fedkiw

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