Decay Rates of the Inverse of Nonsymmetric Tridiagonal and Band Matrices

Abstract

It is well known that the inverse C = [ci,j] of an irreducible nonsingular symmetric tridiagonal matrix is given by two sequences of real numbers, {ui} and {vi}, such that ci,j = u i vj for $i \leq j$. A similar result holds for nonsymmetric matrices A. There the inverse can be described by four sequences {ui},{vi}, {xi},$ and {vi} with u ivi = xiyi. Here we characterize certain properties of A, i.e., being an M-matrix or positive definite, in terms of the ui, vi,xi, and yi. We also establish a relation of zero row sums and zero column sums of A and pairwise constant ui,vi, xi, and yi. Moreover, we consider decay rates for the entries of the inverse of tridiagonal and block tridiagonal (banded) matrices. For diagonally dominant matrices we show that the entries of the inverse strictly decay along a row or column. We give a sharp decay result for tridiagonal irreducible M-matrices and tridiagonal positive definite matrices. We also give a decay rate for arbitrary banded M-matrices.