Spaces of triangularizable matrices
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866912323305734144 |
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| author | Pazzis, Clément de Seguins |
| author_facet | Pazzis, Clément de Seguins |
| contents | Let F be a field. We investigate the greatest possible dimension t_n(F) for a vector space of n-by-n matrices with entries in F and in which every element is triangularizable over the ground field F. It is obvious that t_n(F) is greater than or equal to n(n+1)/2, and we prove that equality holds if and only if F is not quadratically closed or n=1, excluding finite fields with characteristic 2. If F is infinite and not quadratically closed, we give an explicit description of the solutions with the critical dimension t_n(F), reducing the problem to the one of deciding for which integers k between 2 and n all k-by-k symmetric matrices over F are triangularizable. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_07942 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Spaces of triangularizable matrices Pazzis, Clément de Seguins Rings and Algebras 15A30, 15A03, 15A18 Let F be a field. We investigate the greatest possible dimension t_n(F) for a vector space of n-by-n matrices with entries in F and in which every element is triangularizable over the ground field F. It is obvious that t_n(F) is greater than or equal to n(n+1)/2, and we prove that equality holds if and only if F is not quadratically closed or n=1, excluding finite fields with characteristic 2. If F is infinite and not quadratically closed, we give an explicit description of the solutions with the critical dimension t_n(F), reducing the problem to the one of deciding for which integers k between 2 and n all k-by-k symmetric matrices over F are triangularizable. |
| title | Spaces of triangularizable matrices |
| topic | Rings and Algebras 15A30, 15A03, 15A18 |
| url | https://arxiv.org/abs/2410.07942 |