Schur complements of generally diagonally dominant matrices and a criterion for irreducibility of matrices

Abstract

As is well known, the Schur complements of strictly or irreducibly diagonally dominant matrices are H−matrices; however, the same is not true of generally diagonally dominant matrices. This paper proposes some conditions on the generally diagonally dominant matrix A and the subset α ⊂{ 1,2,...,n} so that the Schur complement matrix A/α is an H−matrix. These conditions are then applied to decide whether a matrix is irreducible or not.