Fifty Three Matrix Factorizations: A systematic approach

TL;DR

This paper systematically catalogs 53 matrix factorizations inspired by Lie theory, unifying known factorizations and introducing new ones, with potential for advancing computational algorithms and applications.

Contribution

It introduces a comprehensive, Lie theory-inspired framework that unifies and extends matrix factorizations, including many previously unknown forms.

Findings

Cataloged 53 matrix factorizations, many new

Unified known factorizations under a systematic approach

Suggested potential for new algorithms and applications

Abstract

The success of matrix factorizations such as the singular value decomposition (SVD) has motivated the search for even more factorizations. We catalog 53 matrix factorizations, most of which we believe to be new. Our systematic approach, inspired by the generalized Cartan decomposition of Lie theory, also encompasses known factorizations such as the SVD, the symmetric eigendecomposition, the CS decomposition, the hyperbolic SVD, structured SVDs, the Takagi factorization, and others thereby covering familiar matrix factorizations as well as ones that were waiting to be discovered. We suggest that Lie theory has one way or another been lurking hidden in the foundations of the very successful field of matrix computations with applications routinely used in so many areas of computation. In this paper, we investigate consequences of the Cartan decomposition and the little known generalized Cartan decomposition for matrix factorizations. We believe that these factorizations once properly identified can lead to further work on algorithmic computations and applications.