Notes About Singular Value Decomposition

A brief summary of SVD:

An original matrix A_mn is represented as a muliplication of three matrices:

    A_mn = U_mmS_mnV_nn^T

The columns of U are the orthonormal engenvectors of AA^T descendingly ordered by the corresponding eigenvalues, and the columns of V are the orthonormal engenvectors of A^TA descendingly ordered by the corresponding eigenvalues. This also suggests that U and V are orthonormal (orthogonal) matrices. A characteristic of the two matrices is that the non-zero eigenvalues of U and V are always the same. S is a diagonal matrix containing the square roots of eigenvalues from U and V in descending order. The diagonal entries in S are the singular values of A, the columns in U are called left singular vectors, and the columns in V are called right singular vectors.

Notes About Singular Value Decomposition

原文：http://www.cnblogs.com/waytofall/p/4560632.html