Characterization of diagonally dominant H-matrices

TL;DR

This paper proves the equivalence of existing sufficient conditions for diagonally dominant matrices to be nonsingular H-matrices and simplifies their characterization, establishing the necessity of these conditions.

Contribution

It unifies and simplifies the criteria for diagonally dominant H-matrices and proves the necessity of previously known sufficient conditions.

Findings

Sufficient conditions for diagonally dominant matrices to be nonsingular H-matrices are equivalent.

Simplified characterization of diagonally dominant H-matrices.

Necessary and sufficient conditions established for these matrices.

Abstract

We first show that sufficient conditions for a diagonally dominant matrix to be a nonsingular one (and also an H-matrix), obtained independently by Shivakumar and Chew in 1974, and Farid in 1995, are equivalent. Then we simplify the characterization of diagonally dominant H-matrices obtained by Huang in 1995, and using it prove that the Shivakumar-Chew-Farid sufficient condition for a diagonally dominant matrix to be an H-matrix, is also necessary.