Necessary and Sufficient Criterion for Singular or Nonsingular of Diagonally Dominant Matrices
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866912738127642624 |
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| author | Jin, Jidong |
| author_facet | Jin, Jidong |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_00463 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Necessary and Sufficient Criterion for Singular or Nonsingular of Diagonally Dominant Matrices Jin, Jidong Rings and Algebras Numerical Analysis Dynamical Systems 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. |
| title | Necessary and Sufficient Criterion for Singular or Nonsingular of Diagonally Dominant Matrices |
| topic | Rings and Algebras Numerical Analysis Dynamical Systems |
| url | https://arxiv.org/abs/2512.00463 |