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