Localization of Matrix Factorizations

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 of when and under what conditions the matrix factors inherit the localization of the original matrix for such fundamental matrix factorizations as the LU, QR, Cholesky, and polar factorizations.