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4 Singular Value Decomposition (SVD) - Princeton University

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4 Singular Value Decomposition (SVD)The Singular Value Decomposition of a matrixAis the factorization ofAinto theproduct of three matricesA=UDVTwhere the columns ofUandVare orthonormaland the matrixDis diagonal with positive real entries. The SVD is useful in many we mention two examples. First, the rank of a matrixAcan be read offfrom itsSVD. This is useful when the elements of the matrix are real numbers that have beenrounded to some finite precision. Before the entries were rounded the matrix may havebeen of low rank but the rounding converted the matrix to full rank. The original rankcan be determined by the number of diagonal elements ofDnot exceedingly close to , for a square and invertible matrixA,theinverseofAisVD gain insight into the SVD, treat the rows of ann dmatrixAasnpoints in ad-dimensional space and consider the problem of finding the bestk-dimensional subspacewith respect to the set of points.

4 Singular Value Decomposition (SVD) The singular value decomposition of a matrix A is the factorization of A into the product of three matrices A = UDVT where the columns of U and V are orthonormal and the matrix D is diagonal with positive real entries. The SVD is useful in many tasks. Here we mention two examples.

  Value, Singular, Decomposition, Singular value decomposition

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