Generalizations of the singular value and QR decompositions

Abstract

In this paper, we provide a state-of-the-art survey of a recently discovered set of generalizations of the ordinary singular value decomposition, which contains all existing generalizations for 2 matrices (such as the product SVD and the quotient SVD) and for 3 matrices (such as the restricted SVD), as special cases. We present the main theorem and a discussion on the structural properties of these generalized singular value decompositions. A proposal for a standardized nomenclature is made as well. At the same time, we summarize some recent results on a corresponding generalization for any number of matrices of the QR (or URV) decomposition.