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
The matrices in this factorization have the following properties:

U [m × n] and V [n × n]

are orthogonal matrices. The columns u_{i} of U =[u_{1}, ..., u_{n}] are the left singular vectors, and the columns v_{i} of V [v_{1}, ..., v_{n}] are the right singular vectors.

Σ [n × n] = diag (σ_{1}, ..., σ_{n})

is a real, nonnegative, and diagonal matrix. Its diagonal contains the so called singular values σ_{i}, where σ_{1} ≥ ... ≥ σ_{n} ≥ 0.
If m < n, the decomposition results in U [m × m] and V [n × m].
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© djmw, May 10, 2012