Decompositions of Matrices Into a Sum of Torsion Matrices and Matrices of Fixed Nilpotence
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| Format: | Preprint |
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2023
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| author | Danchev, Peter García, Esther Lozano, Miguel Gómez |
| author_facet | Danchev, Peter García, Esther Lozano, Miguel Gómez |
| contents | For $n\ge 2$ and fixed $k\ge 1$, we study when a square matrix $A$ over an arbitrary field $\mathbb{F}$ can be decomposed as $T+N$ where $T$ is a torsion matrix and $N$ is a nilpotent matrix with $N^k=0$. For fields of prime characteristic, we show that this decomposition holds as soon as the characteristic polynomial of $A\in \mathbb{M}_{n}(\mathbb{F})$ is algebraic over its base field and the rank of $A$ is at least $\frac nk$, and we present several examples that show that the decomposition does not hold in general. Furthermore, we completely solve this decomposition problem for $k=2$ and nilpotent matrices over arbitrary fields (even over division rings). This somewhat continues our recent publications in Lin. \& Multilin. Algebra (2023) and Internat. J. Algebra \& Computat. (2022) as well as it strengthens results due to Calugareanu-Lam in J. Algebra \& Appl. (2016). |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2301_06106 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Decompositions of Matrices Into a Sum of Torsion Matrices and Matrices of Fixed Nilpotence Danchev, Peter García, Esther Lozano, Miguel Gómez Rings and Algebras 15A21, 15A24, 15B99, 16U60 For $n\ge 2$ and fixed $k\ge 1$, we study when a square matrix $A$ over an arbitrary field $\mathbb{F}$ can be decomposed as $T+N$ where $T$ is a torsion matrix and $N$ is a nilpotent matrix with $N^k=0$. For fields of prime characteristic, we show that this decomposition holds as soon as the characteristic polynomial of $A\in \mathbb{M}_{n}(\mathbb{F})$ is algebraic over its base field and the rank of $A$ is at least $\frac nk$, and we present several examples that show that the decomposition does not hold in general. Furthermore, we completely solve this decomposition problem for $k=2$ and nilpotent matrices over arbitrary fields (even over division rings). This somewhat continues our recent publications in Lin. \& Multilin. Algebra (2023) and Internat. J. Algebra \& Computat. (2022) as well as it strengthens results due to Calugareanu-Lam in J. Algebra \& Appl. (2016). |
| title | Decompositions of Matrices Into a Sum of Torsion Matrices and Matrices of Fixed Nilpotence |
| topic | Rings and Algebras 15A21, 15A24, 15B99, 16U60 |
| url | https://arxiv.org/abs/2301.06106 |