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

TL;DR

This paper investigates conditions under which matrices can be decomposed into a sum of a torsion matrix and a nilpotent matrix of fixed index, providing complete solutions for certain cases and counterexamples for others.

Contribution

It offers a comprehensive analysis of matrix decompositions into torsion and nilpotent parts, including new results for fields of prime characteristic and specific cases like k=2.

Findings

Decomposition holds under certain rank and characteristic polynomial conditions.

Complete solution for k=2 and nilpotent matrices over arbitrary fields.

Counterexamples show the decomposition does not always exist.

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).