A tutorial on the singular value decomposition

An m x n real matrix A can be factored as; $UWV^T$ y where U and V are orthonormal, and W is upper left diagonal. This factorization is c&lled Singular Value Decomposition (SVDJ. The matrices U, W, and V are useful in characterizing the matrix A. In this manuscript geometric characterizations are emphasized. Geometric characterizations are analyzed in terms of subspaces, matrix scaling, and norms. We also present a numerical viewpoint for SVD in order to keep the material self-contained. In the last section we treat a special problem where action of the matrix A is restricted to a given subspace.