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Singular Value Decomposition Raul Queiroz Feitosa 25/5/2010 Sigular Value Decomposiion 2 Definition Any m×n matrix A can be written as: A = U D V T where � the columns of the m×m matrix U are mutually orthogonal unity vectors, � as well as the columns of n×n matrix V, and � the m×n matrix D is diagonal, its elements σi (i=1,2,...,n) are called (σ1 ≥ σ2 ≥… σn ≥0 ) singular values. 25/5/2010 Sigular Value Decomposiion 3 Example = 00040 00000 00300 20001 A − = 0001 1000 0010 0100 U = 00000 00500 00030 00004 D − = 2.00008.0 01000 8.00002.0 00100 00010 V A = U D V T 25/5/2010 Sigular Value Decomposiion 4 Properties � A square matrix A is nonsingular if and only if all its singular values are different from zero. � The ratio C= σ1/ σn, called condition number, measures the degree of singularity of A. When 1/C is comparable to the arithmetic precision of the machine, the matrix A is ill-conditioned and, for all practical purposes, can be considered singular. 25/5/2010 Sigular Value Decomposiion 5 Properties (cont.) � The squares of the non zero singular values are the nonzero eigenvalues of both the n×n matrix AAT and m×m matrix ATA. � The columns of U are the eigenvectors of AAT � The columns of V are the eigenvectors of ATA. 25/5/2010 Sigular Value Decomposiion 6 Singular Value Decomposition END
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