#### Vol. 13, No. 2, 2020

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Enumerating diagonalizable matrices over $\mathbb{Z}_{p^k}$

### Catherine Falvey, Heewon Hah, William Sheppard, Brian Sittinger and Rico Vicente

Vol. 13 (2020), No. 2, 323–344
##### Abstract

Although a good portion of elementary linear algebra concerns itself with matrices over a field such as $ℝ$ or $ℂ$, 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 ${ℤ}_{{p}^{k}}$ that are diagonalizable over ${ℤ}_{{p}^{k}}$. This turns out to be significantly more nontrivial than its finite-field counterpart due to the presence of zero divisors in ${ℤ}_{{p}^{k}}$.

##### Keywords
eigenvalues, matrices, finite commutative rings
##### Mathematical Subject Classification 2010
Primary: 05A05, 05C22, 15A18, 15B33