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

TL;DR

The paper tackles the classical problem of when diagonally dominant matrices are singular or nonsingular by building on Taussky’s theorem and reducing reducible matrices to irreducible Frobenius blocks. It develops a comprehensive similarity and unitary similarity framework for singular irreducible diagonally dominant matrices, culminating in a necessary-and-sufficient condition expressed as a consistent system of angle equations for the nonzero off-diagonal entries. A key outcome is that singularity of irreducible diagonally dominant matrices is characterized by both weak diagonal dominance and the solvability of the angle equations, enabling a concrete decision procedure. The analysis also connects these matrices to real doubly balanced forms and Markov-type structures, with clear applications to network dynamics and Kolmogorov 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.