Some new results on generalized diagonally dominant matrices and matrix eigenvalue inclusion regions

TL;DR

This paper introduces new classes of G-function pairs to improve eigenvalue inclusion regions and conditions for generalized diagonally dominant matrices, extending classical results in matrix theory.

Contribution

It proposes a class of G-function pairs extending existing concepts, establishing new conditions and regions for eigenvalues and diagonal dominance.

Findings

New G-function pairs are established and characterized.

Sufficient and necessary conditions for diagonal dominance are derived.

Enhanced eigenvalue inclusion regions outperform classical results.

Abstract

In matrix theory and numerical analysis there are two very famous and important results. One is Gersgorin circle theorem, the other is strictly diagonally dominant theorem. They have important application and research value, and have been widely used and studied. In this paper, we investigate generalized diagonally dominant matrices and matrix eigenvalue inclusion regions. A class of G-function pairs is proposed, which extends the concept of G-functions. Thirteen kind of G-function pairs are established. Their properties and characteristics are studied. By using these special G-function pairs, we construct a large number of sufficient and necessary conditions for strictly diagonally dominant matrices and matrix eigenvalue inclusion regions. These conditions and regions are composed of different combinations of G-function pairs, deleted absolute row sums and column sums of matrices. The results extend, include and are better than some classical results.