Singular M-matrices which may not have a nonnegative generalized inverse

A matrix A ∈ ℝn×n is a GM-matrix if A = sI − B, where 0 < ρ(B) ≤ s and B ∈WPFn i.e., both B and Bthave ρ(B) as their eigenvalues and their corresponding eigenvector is entry wise nonnegative. In this article,we consider a generalization of a subclass of GM-matrices having a nonnegative core nilpotent decompositionand prove a characterization result for such matrices. Also, we study various notions of splitting of matricesfrom this new class and obtain sufficient conditions for their convergence.