singular value decomposition

The singular value decomposition (SVD) is a matrix factorization algorithm.

For m > n, the singular value decomposition of a real m × n matrix A is the factorization

A = U Σ V′,

The matrices in this factorization have the following properties:

U [m × n] and V [n × n]
are orthogonal matrices. The columns ui of U =[u1, ..., un] are the left singular vectors, and the columns vi of V [v1, ..., vn] are the right singular vectors.
Σ [n × n] = diag (σ1, ..., σn)
is a real, nonnegative, and diagonal matrix. Its diagonal contains the so called singular values σi, where σ1 ≥ ... ≥ σn ≥ 0.

If m < n, the decomposition results in U [m × m] and V [n × m].

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© djmw, May 10, 2012