Matrices with banded inverses: inversion algorithms and factorization of Gauss-Markov processes

Abstract

The paper considers the inversion of full matrices whose inverses are L-banded. We derive a nested inversion algorithm for such matrices. Applied to a tridiagonal matrix, the algorithm provides its explicit inverse as an element-wise product (Hadamard product) of three matrices. When related to Gauss-Markov random processes (GMrp), this result provides a closed-form factored expression for the covariance matrix of a first-order GMrp. This factored form leads to the interpretation of a first-order GMrp as the product of three independent processes: a forward independent-increments process, a backward independent-increments process, and a variance-stationary process. We explore the nonuniqueness of the factorization and design it so that the forward and backward factor processes have minimum energy. We then consider the issue of approximating general nonstationary Gaussian processes by Gauss-Markov processes under two optimality criteria: the Kullback-Leibler distance and maximum entropy. The problem reduces to approximating general covariances by covariance matrices whose inverses are banded. Our inversion result is an efficient algorithmic solution to this problem. We evaluate the information loss between the original process and its Gauss-Markov approximation.