Doubly diagonally dominant matrices

We consider the class of doubly diagonally dominant matrices (A = [aij] ε Cn, n, aiiajj ∑k ≠ i aik∑k jajk, i ≠ j) and its subclasses. We give necessary and sufficient conditions in terms of the directed graph for an irreducibly doubly diagonally dominant matrix to be a singular matrix or to be an H-matrix. As in the case of diagonal dominance, we show that the Schur complements of doubly diagonally dominant matrices inherit this property. Moreover, we describe when a Schur complement of a strictly doubly diagonally dominant matrix is strictly diagonally dominant.