Monotonicity of Perturbed Tridiagonal $M$-Matrices

Abstract

A well-known property of an $M$-matrix is that its inverse is elementwise nonnegative, which we write as $M^{-1} \geq 0$. In a previous paper [Linear Algebra Appl., 434 (2011), pp. 131--143], we gave sufficient bounds on single element perturbations so that monotonicity persists for a perturbed tridiagonal $M$-matrix. Here we extend these results, presenting the actual maximum upper bounds on single element perturbations, as well as sufficient and necessary conditions for the maximum allowable higher rank perturbations. Perturbed Toeplitz tridiagonal $M$-matrices are considered as a special case. We compare our results to existing normwise bounds due to Bouchon and an iterative algorithm provided by Buffoni. We demonstrate the utility of these results by considering an application: ensuring a nonnegative solution of a discrete analogue of an integro-differential population model.