Singular Values
Singular values are
Contents
Introduction
Any linear transformation can be rewritten as a rotation (i.e., from the original basis onto a convenient basis), a scaling and stretching, and another rotation (i.e., from the convenient basis onto the destination basis).
Therefore A = UΣVT, where the above descriptions correspond to V, Σ, and U in that order.
Description
Singular values are the square roots of eigenvalues. Specifically, a matrix A has singular values (σ) corresponding to the eigenvalues of ATA.
Matrices of all sizes have singular values. For a matrix of size m x n, there are at most min{m,n} singular values. Furthermore, the number of non-zero singular values is equal to the rank of A.
These singular values are the numbers on the diagonal of Σ. Naturally Σ matches A as a m x n matrix.