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Singular value decomposition

Suppose we have a rectangular matrix A with m rows and n columns, and that p is the minimum of m and n. The singular value decomposition can be defined as

mAn = mUp pSp pVn

The diagonal matrix S contains the p singular values of A, ordered such that

s1s2 ≥ … ≥ sp ≥ 0

The matrices U and V contain the left and right singular vectors of A, and are orthonormal: that is,

U′U = V′V = Ip

The smaller of U and V will be orthogonal. So, if A has more rows than columns, m>n, p=n and VV′=Ip.

Updated on December 4, 2017

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