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arXiv · 2301.06106

Decompositions of Matrices Into a Sum of Torsion Matrices and Matrices of Fixed Nilpotence

Peter Danchev·Esther García·Miguel Gómez Lozano
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Abstract

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).

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Peter Danchev, Esther García, Miguel Gómez Lozano. 2023-01-15. Decompositions of Matrices Into a Sum of Torsion Matrices and Matrices of Fixed Nilpotence. https://doi.org/10.1016/j.laa.2023.07.005

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