A tree of generalizations of the ordinary singular value decomposition

Abstract

It is shown how to generalize the ordinary singular value decomposition of a matrix into a combined factorization of any number of matrices. We propose to call these factorizations generalized singular value decompositions. For two matrices, this reduces to the product and quotient singular value decompositions. One of the factorizations for three matrices is the restricted singular value decomposition. These generalizations form a tree of factorizations, where at level k, for k matrices, there are 2 k factorizations, not all of which are independent. The different levels are related to each other in a recursive fashion. Any generalized singular value decomposition for k matrices can be constructed from a decomposition for k – 1 matrices. This results in an inductive proof which uses only the ordinary singular value decomposition. Several examples are analysed in detail.