Sums of two nilpotent quaternionic matrices
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915466070458368 |
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| author | Pazzis, Clément de Seguins |
| author_facet | Pazzis, Clément de Seguins |
| contents | Let $\mathcal{Q}$ be a quaternion division algebra over a field, and $n \geq 2$ be an integer. In a recent article, de La Cruz et al have proved that every $n$-by-$n$ matrix with entries in $\mathcal{Q}$ and pure quaternionic trace is the sum of three nilpotent matrices, and they have shown that some are not the sum of two nilpotent matrices.
Here, we give a simple characterization of the square matrices with entries in $\mathcal{Q}$ that are the sum of two nilpotent ones. When $n \geq 3$, the special cases involve the scalar matrices and their perturbations by rank $1$ matrices, as well as the very special case of $3$-by-$3$ unispectral diagonalisable matrices. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_19627 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Sums of two nilpotent quaternionic matrices Pazzis, Clément de Seguins Rings and Algebras 15B33, 15A23, 15A18 Let $\mathcal{Q}$ be a quaternion division algebra over a field, and $n \geq 2$ be an integer. In a recent article, de La Cruz et al have proved that every $n$-by-$n$ matrix with entries in $\mathcal{Q}$ and pure quaternionic trace is the sum of three nilpotent matrices, and they have shown that some are not the sum of two nilpotent matrices. Here, we give a simple characterization of the square matrices with entries in $\mathcal{Q}$ that are the sum of two nilpotent ones. When $n \geq 3$, the special cases involve the scalar matrices and their perturbations by rank $1$ matrices, as well as the very special case of $3$-by-$3$ unispectral diagonalisable matrices. |
| title | Sums of two nilpotent quaternionic matrices |
| topic | Rings and Algebras 15B33, 15A23, 15A18 |
| url | https://arxiv.org/abs/2508.19627 |