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

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.