Localization of Matrix Factorizations

TL;DR

This paper develops a comprehensive theoretical framework to determine when and how matrix factorizations preserve the off-diagonal decay properties of matrices, which is crucial in various scientific and engineering applications.

Contribution

It provides the first general theory establishing conditions under which LU, QR, Cholesky, and polar factorizations inherit matrix localization properties.

Findings

Framework for inheritance of decay properties in factorizations

Conditions identified for LU, QR, Cholesky, and polar factorizations

Applicable to matrices with off-diagonal decay in multiple fields

Abstract

Matrices with off-diagonal decay appear in a variety of fields in mathematics and in numerous applications, such as signal processing, statistics, communications engineering, condensed matter physics, and quantum chemistry. Numerical algorithms dealing with such matrices often take advantage (implicitly or explicitly) of the empirical observation that this off-diagonal decay property seems to be preserved when computing various useful matrix factorizations, such as the Cholesky factorization or the QR-factorization. There is a fairly extensive theory describing when the inverse of a matrix inherits the localization properties of the original matrix. Yet, except for the special case of band matrices, surprisingly very little theory exists that would establish similar results for matrix factorizations. We will derive a comprehensive framework to rigorously answer the question when and under which conditions the matrix factors inherit the localization of the original matrix for such fundamental matrix factorizations as the LU-, QR-, Cholesky, and Polar factorization.