M-matrices leading to semiconvergent splittings (Q1239214)

An M-matrix as defined by Ostrowski is a matrix that can be split into \(A =sI- B\), \(s>0\), \(B\geq \rho (B)\), the spectral radius of \(B\). M-matrices with the property that the powers of \(T=(1/s)B\) converge for some \(s\) are studied and are characterized here in terms of the nonnegativity of the group generalized inverse of \(A\) on the range space of \(A\), extending the well-known property that \(A^{-1}\geq 0\) whenever \(A\) is nonsingular. Extensions and applications of these new concepts may be found in \textit{C. D. Meyer, jun.} and \textit{R. J. Plemmons} [SIAM J. Numer. Anal. 14, 699--705 (1977; Zbl 0366.65017)].