Matrices over Finite Fields of Characteristic 2 as Sums of Diagonalizable and Square-Zero Matrices
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866910136082104320 |
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| author | Danchev, Peter García, Esther Lozano, Miguel Gómez |
| author_facet | Danchev, Peter García, Esther Lozano, Miguel Gómez |
| contents | We investigate the problem asking when any square matrix whose entries lie in a finite field of characteristic 2 is decomposable into the sum of a diagonalizable matrix and a nilpotent matrix with index of nilpotency at most 2 and, as a result, we completely resolve this question in the affirmative for any finite field of characteristic 2 having strictly more than three elements. 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 Šter in Linear Algebra & Appl. (2018) and Shitov in Indag. Math. (2019) (see, respectively, [9] and [8]). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_15286 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Matrices over Finite Fields of Characteristic 2 as Sums of Diagonalizable and Square-Zero Matrices Danchev, Peter García, Esther Lozano, Miguel Gómez Rings and Algebras 15A15, 15A21, 15A83 We investigate the problem asking when any square matrix whose entries lie in a finite field of characteristic 2 is decomposable into the sum of a diagonalizable matrix and a nilpotent matrix with index of nilpotency at most 2 and, as a result, we completely resolve this question in the affirmative for any finite field of characteristic 2 having strictly more than three elements. 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 Šter in Linear Algebra & Appl. (2018) and Shitov in Indag. Math. (2019) (see, respectively, [9] and [8]). |
| title | Matrices over Finite Fields of Characteristic 2 as Sums of Diagonalizable and Square-Zero Matrices |
| topic | Rings and Algebras 15A15, 15A21, 15A83 |
| url | https://arxiv.org/abs/2604.15286 |