An index classification of M-matrices

An M-matrix as defined by Ostrowski 5] is a matrix that can be split into A = sI ? B, where s > 0, B ? 0, with s ? r(B), the spectral radius of B. Following Plemmons 6], we develop a classification of all M-matrices. We consider v, the index of zero for A, i.e., the smallest nonnegative integer n such that the null spaces of Aⁿ and Aⁿ⁺¹ coincide. We characterize this index in terms of convergence properties of powers of s^?1B. We develop additional characterizations in terms of nonnegativity of the Drazin inverse of A on the range of A^v, extending (as conjectured by Poole and Boullion 7]) the well-known property that A^?1?0 whenever A is nonsingular.