有限域特征为 2 的矩阵可表示为对角化矩阵与平方零矩阵之和

我们研究了当任何方阵的条目位于特征为 2 的有限域时，是否可以分解为具有至多指数为 2 的零化数矩阵之和以及至多指数为 2 的零化数矩阵的问题。我们在特征为 2 且元素数严格大于 3 的所有有限域上完全以肯定方式解决了这个问题。Our main theorem of that type, combined with results from our recent publication in Linear Algebra & Appl. (2026) (see [7]), totally settle this problem for all finite fields different from $\mathbb{F}_2$ and $\mathbb{F}_3$. However, in this paper we also prove that each matrix over $\mathbb{F}_2$ is expressible as the sum of a potent matrix with index of potency not exceeding 4 and a nilpotent matrix with index of nilpotency not exceeding 2, thus substantiating recent examples due to \v{S}ter in Linear Algebra & Appl. (2018) and Shitov in Indag. Math. (2019) (see, respectively, [9] and [8]).