Necessary and Sufficient Criterion for Singular or Nonsingular of Diagonally Dominant Matrices

TL;DR

This paper provides a complete set of necessary and sufficient conditions to determine when diagonally dominant matrices are singular or nonsingular, extending classical matrix theory with new theoretical insights.

Contribution

It develops a unified theoretical framework for diagonally dominant matrices, including a similarity analysis and an angle equation system for irreducible cases, advancing the understanding of matrix singularity.

Findings

Established necessary and sufficient conditions for singularity of diagonally dominant matrices.

Reduced the problem to irreducible matrices and analyzed their properties.

Connected matrix theory results to applications in network systems and differential equations.

Abstract

The problem of determining whether a diagonally dominant matrix is singular or nonsingular is a classical topic in matrix theory. This paper develops necessary and sufficient conditions for the singularity or nonsingularity of diagonally dominant matrices. Starting from Taussky's theorem, we establish a unified line of theory which reduces the general problem to the study of irreducible diagonally dominant matrices. A complete similarity and unitary similarity analysis is given for singular irreducible diagonally dominant matrices, leading to a necessary and sufficient condition expressed in terms of an angle equation system associated with the nonzero off-diagonal entries. Applications and motivations from network dynamical systems, affine multi-agent systems, and Kolmogorov differential equations are also discussed.