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We present a test for determining if a substochastic matrix is convergent. Byestablishing a duality between weakly chained diagonally dominant (w.c.d.d.)L-matrices and convergent substochastic matrices, we show that this test can betrivially 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 ofthe input matrix if it is sparse and quadratic if it is dense. This is apartial strengthening of the cubic test in [J. M. Pe\~na., A stable test tocheck 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 atleast 1964. We point out that this strengthens some recent results onM-matrices in the literature.

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