Similarity of matrices over local rings of length two

TL;DR

This paper develops a normal form theory for similarity classes of matrices over local rings of length two, providing explicit classifications, enumeration, and properties of these classes, including their relation to transposes.

Contribution

It introduces a new framework for classifying similarity classes in matrix rings over certain local rings using module extensions, and enumerates these classes with polynomial formulas.

Findings

Classifications of similarity classes for n ≤ 4

Enumeration formulas for classes over finite fields

Existence of non-self-transpose classes for all n > 3

Abstract

Let $R$ be a local principal ideal ring of length two, for example, the ring $R=\Z/p^2\Z$ with $p$ prime. In this paper we develop a theory of normal forms for similarity classes in the matrix rings $M_n(R)$ by interpreting them in terms of extensions of $R[t]$-modules. Using this theory, we describe the similarity classes in $M_n(R)$ for $n\leq 4$, along with their centralizers. Among these, we characterize those classes which are similar to their transposes. Non-self-transpose classes are shown to exist for all $n>3$. When $R$ has finite residue field of order $q$, we enumerate the similarity classes and the cardinalities of their centralizers as polynomials in $q$. Surprisingly, the polynomials representing the number of similarity classes in $M_n(R)$ turn out to have non-negative integer coefficients.