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

TL;DR

This paper investigates the enumeration of diagonalizable matrices over the ring _{p^k}, highlighting the complexities introduced by zero divisors compared to finite fields.

Contribution

It provides a novel enumeration approach for diagonalizable matrices over _{p^k}, addressing challenges posed by zero divisors.

Findings

Enumeration formulas for diagonalizable matrices over _{p^k}

Analysis of zero divisors impact on diagonalizability

Comparison with finite field cases

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