ABSTRACT

In order to find a suitable expression of an arbitrary square matrix over an arbitrary field, we prove that that every square matrix over an infinite field is always representable as a sum of a diagonalizable matrix and a nilpotent matrix of order less than or equal to two. In addition, each $2\times 2$ matrix over any field admits such a representation. We, moreover, show that, for all natural numbers $n\ge 3$, every $n\times n$ matrix over a finite field having no less than n + 1 elements also admits such a decomposition. The latter completes a recent example due to Breaz [Matrices over finite fields as sums of periodic and nilpotent elements. Linear Algebra Appl. 2018;555:92–97]. As a consequence of these decompositions, we show that every nilpotent matrix over a field can be expressed as the sum of a potent matrix and a square-zero matrix. This somewhat improves on recent results due to Abyzov et al. [On some matrix analogues of the little Fermat theorem. Mat Zametki. 2017;101(2):163–168] and Shitov [The ring ${\mathbb{M}}_{8k+4}\left({\mathbb{Z}}_2\right)$ is nil-clean of index four. Indag Math (N.S.). 2019;30:1077–1078].

摘要

为了找到任意场上任意平方矩阵的合适表达式，我们证明了无限场上的每个平方矩阵总是可表示为可对角矩阵和阶小于或等于2的幂等矩阵之和。 。另外，每个$2\times 2$任何字段上的矩阵都允许这种表示。此外，我们证明对于所有自然数$ñ\ge 3$，每个 $ñ\times ñ$具有不小于n  +1个元素的有限域上的矩阵也允许这种分解。后者由于Breaz [有限域上的矩阵作为周期元素和幂等元素之和而完成了一个最近的例子。线性代数应用 2018; 555：92–97]。这些分解的结果是，我们证明了一个场上的每个幂等矩阵都可以表示为一个有效矩阵和一个零平方矩阵之和。由于Abyzov等人的最新研究结果，这有所改善。[关于小费马定理的某些矩阵类似物。Mat Zametki。2017; 101（2）：163–168]和Shitov [指环${\mathbb{中号}}_{8ķ+4}（{\mathbb{ž}}_2）$指数为零。Indag数学（NS）。2019; 30：1077-1078]。