Inverse positivity of perturbed tridiagonal \(M\)-matrices (Q1014498)

A matrix M is an \(M\)-matrix if and only if \(m_{ij} \leq 0\) for \(i \neq j\), \(m_{ii}>0\) and M is generalized strictly diagonally dominant. If M is \(M\)-matrix then it is non-singular and M\(^{-1} \geq 0\), where the inequality is satisfied element-wise. In the paper, the inverses of perturbed \(M\)-matrices are considered. When a general element of a tridiagonal \(M\)-matrix is perturbed then sufficient upper bounds on the size of this perturbation were found which ensure that the inverse of the perturbed matrix is non-negative. Bounds on the maximum allowable perturbations made to a general element of a non-symmetric tridiagonal \(M\)-matrix are also given. The analogous results are obtained for symmetric and Toeplitz matrices as special cases.