Decompositions of Periodic Matrices into a Sum of Special Matrices
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866908971456004096 |
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| author | Danchev, Peter García, Esther Lozano, Miguel Gómez |
| author_facet | Danchev, Peter García, Esther Lozano, Miguel Gómez |
| contents | We study the problem of when a periodic square matrix of order $n$ over an arbitrary field $\mathbb{F}$ is decomposable into the sum of a square-zero matrix and a torsion matrix, and show that this decomposition can always be obtained for matrices of rank at least $\frac n2$ when $\mathbb{F}$ is either a field of prime characteristic, or the field of rational numbers, or an algebraically closed field of zero characteristic. We also provide a counterexample to such a decomposition when $\mathbb{F}$ equals the field of the real numbers. Moreover, we prove that each periodic square matrix over any field is a sum of an idempotent matrix and a torsion matrix. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_02777 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Decompositions of Periodic Matrices into a Sum of Special Matrices Danchev, Peter García, Esther Lozano, Miguel Gómez Rings and Algebras 15A21, 15A24, 16U60 We study the problem of when a periodic square matrix of order $n$ over an arbitrary field $\mathbb{F}$ is decomposable into the sum of a square-zero matrix and a torsion matrix, and show that this decomposition can always be obtained for matrices of rank at least $\frac n2$ when $\mathbb{F}$ is either a field of prime characteristic, or the field of rational numbers, or an algebraically closed field of zero characteristic. We also provide a counterexample to such a decomposition when $\mathbb{F}$ equals the field of the real numbers. Moreover, we prove that each periodic square matrix over any field is a sum of an idempotent matrix and a torsion matrix. |
| title | Decompositions of Periodic Matrices into a Sum of Special Matrices |
| topic | Rings and Algebras 15A21, 15A24, 16U60 |
| url | https://arxiv.org/abs/2404.02777 |