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

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