Singular Value Decomposition ©Copyright 2011 Herbert J. Bernstein A vector x is an eigenvector of a matrix M with eigenvalue λ if

M x = λ x

A real matrix M is called symmetric if M = M^T. The eigenvalues and eigenvectors of a symmetric matrix are readily computed using stable and reliable algorithms. Indeed, such a matrix can be diagonalized by a rotation of the space putting the eigenvalues on the diagional (see http://en.wikipedia.org/wiki/Spectral_theorem. In general a non-symmetric matrix cannot be diagonalized.

In order to get some sense of the spectrum of a general real matrix, M, we derive a symmetric matrix L and R from M by

L = MM^T R = M^TM

The eigenvalues of L and R are non-negative and form the same set of value. The square roots of those eigenvalues are called the singular values of M