Enumerating Diagonalizable Matrices over $\mathbb{Z}_{p^k}$
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arXiv
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| Main Authors: | , , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866909430160818176 |
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| author | Falvey, Catherine Hah, Heewon Sheppard, William Sittinger, Brian Vicente, Rico |
| author_facet | Falvey, Catherine Hah, Heewon Sheppard, William Sittinger, Brian Vicente, Rico |
| contents | Although a good portion of elementary linear algebra concerns itself with matrices over a field such as $\mathbb{R}$ or $\mathbb{C}$, many combinatorial problems naturally surface when we instead work with matrices over a finite field. As some recent work has been done in these areas, we turn our attention to the problem of enumerating the square matrices with entries in $\mathbb{Z}_{p^k}$ that are diagonalizable over $\mathbb{Z}_{p^k}$. This turns out to be significantly more nontrivial than its finite field counterpart due to the presence of zero divisors in $\mathbb{Z}_{p^k}$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_11358 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Enumerating Diagonalizable Matrices over $\mathbb{Z}_{p^k}$ Falvey, Catherine Hah, Heewon Sheppard, William Sittinger, Brian Vicente, Rico Combinatorics Group Theory Number Theory 05A99, 15B33, 11Z99 Although a good portion of elementary linear algebra concerns itself with matrices over a field such as $\mathbb{R}$ or $\mathbb{C}$, many combinatorial problems naturally surface when we instead work with matrices over a finite field. As some recent work has been done in these areas, we turn our attention to the problem of enumerating the square matrices with entries in $\mathbb{Z}_{p^k}$ that are diagonalizable over $\mathbb{Z}_{p^k}$. This turns out to be significantly more nontrivial than its finite field counterpart due to the presence of zero divisors in $\mathbb{Z}_{p^k}$. |
| title | Enumerating Diagonalizable Matrices over $\mathbb{Z}_{p^k}$ |
| topic | Combinatorics Group Theory Number Theory 05A99, 15B33, 11Z99 |
| url | https://arxiv.org/abs/2412.11358 |