# A fast and stable test to check if a weakly diagonally dominant matrix is a nonsingular M-matrix

## TL;DR

This paper introduces a fast, stable test to determine if a weakly diagonally dominant matrix is a nonsingular M-matrix, improving computational efficiency and strengthening theoretical understanding of M-matrix properties.

## Contribution

The paper extends a test for convergence of substochastic matrices to identify nonsingular M-matrices efficiently, and proves a new property linking nonsingular w.d.d. M-matrices with w.c.d.d. L-matrices.

## Key findings

- The test runs in linear time for sparse matrices.
- The test runs in quadratic time for dense matrices.
- Nonsingular w.d.d. M-matrices are w.c.d.d. L-matrices.

## Abstract

We present a test for determining if a substochastic matrix is convergent. By establishing a duality between weakly chained diagonally dominant (w.c.d.d.) L-matrices and convergent substochastic matrices, we show that this test can be trivially extended to determine whether a weakly diagonally dominant (w.d.d.) matrix is a nonsingular M-matrix. The test's runtime is linear in the order of the input matrix if it is sparse and quadratic if it is dense. This is a partial strengthening of the cubic test in [J. M. Pe\~na., A stable test to check if a matrix is a nonsingular M-matrix, Math. Comp., 247, 1385-1392, 2004]. As a by-product of our analysis, we prove that a nonsingular w.d.d. M-matrix is a w.c.d.d. L-matrix, a fact whose converse has been known since at least 1964. We point out that this strengthens some recent results on M-matrices in the literature.