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arXiv · 2412.11358

Enumerating Diagonalizable Matrices over $\mathbb{Z}_{p^k}$

Catherine Falvey·Heewon Hah·William Sheppard·Brian Sittinger·Rico Vicente
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Abstract

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 $\mathbb{Z}_{p^k}$.

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Catherine Falvey, Heewon Hah, William Sheppard, Brian Sittinger, Rico Vicente. 2024-12-16. Enumerating Diagonalizable Matrices over $\mathbb{Z}_{p^k}$. https://arxiv.org/abs/2412.11358

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