Spaces of triangularizable matrices (II): Finite fields with odd characteristic
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912569703268352 |
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| author | Pazzis, Clément de Seguins |
| author_facet | Pazzis, Clément de Seguins |
| contents | Let $\mathbb{F}$ be a field. Denote by $t_n(\mathbb{F})$ the greatest possible dimension for a vector space of $n$-by-$n$ matrices over $\mathbb{F}$ in which every element is triangularizable over $\mathbb{F}$.
It was recently proved that $t_n(\mathbb{F})=\frac{n(n+1)}{2}$ if and only if $\mathbb{F}$ is not quadratically closed. The structure of the spaces of maximal dimension was also elucidated provided $\mathbb{F}$ is infinite and not quadratically closed. In this sequel, we extend this result to finite fields with odd characteristic. More specifically, we prove that if $\mathbb{F}$ is finite with odd characteristic, then the space of all upper-triangular $n$-by-$n$ matrices is, up to conjugation, the sole vector space of $n$-by-$n$ matrices that has dimension $\frac{n(n+1)}{2}$ and consists only of triangularizable matrices. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_03716 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Spaces of triangularizable matrices (II): Finite fields with odd characteristic Pazzis, Clément de Seguins Rings and Algebras 15A30, 15A03, 15A18 Let $\mathbb{F}$ be a field. Denote by $t_n(\mathbb{F})$ the greatest possible dimension for a vector space of $n$-by-$n$ matrices over $\mathbb{F}$ in which every element is triangularizable over $\mathbb{F}$. It was recently proved that $t_n(\mathbb{F})=\frac{n(n+1)}{2}$ if and only if $\mathbb{F}$ is not quadratically closed. The structure of the spaces of maximal dimension was also elucidated provided $\mathbb{F}$ is infinite and not quadratically closed. In this sequel, we extend this result to finite fields with odd characteristic. More specifically, we prove that if $\mathbb{F}$ is finite with odd characteristic, then the space of all upper-triangular $n$-by-$n$ matrices is, up to conjugation, the sole vector space of $n$-by-$n$ matrices that has dimension $\frac{n(n+1)}{2}$ and consists only of triangularizable matrices. |
| title | Spaces of triangularizable matrices (II): Finite fields with odd characteristic |
| topic | Rings and Algebras 15A30, 15A03, 15A18 |
| url | https://arxiv.org/abs/2509.03716 |