Determinants of some Special Matrices over Commutative Finite Chain Rings

TL;DR

This paper investigates the determinants and enumeration of diagonal and circulant matrices over commutative finite chain rings, providing explicit formulas and exploring their algebraic properties and applications.

Contribution

It offers new formulas for counting invertible and singular circulant matrices over finite chain rings, extending understanding of their algebraic structure and determinants.

Findings

Number of diagonal matrices with a given determinant over R is determined.

Enumeration of nonsingular circulant matrices over R is provided for certain conditions.

Connections between circulant and diagonal matrices are established for counting singular matrices.

Abstract

Circulant matrices over finite fields and over commutative finite chain rings have been of interest due to their nice algebraic structures and wide applications. In many cases, such matrices over rings have a closed connection with diagonal matrices over their extension rings. In this paper, the determinants of diagonal and circulant matrices over commutative finite chain rings $R$ with residue field $\mathbb{F}_q$ are studied. The number of $n\times n$ diagonal matrices over ${R}$ of determinant $a$ is determined for all elements $a$ in $ {R}$ and for all positive integers $n$. Subsequently, the enumeration of nonsingular $n\times n$ circulant matrices over ${R}$ of determinant $a$ is given for all units $a$ in $ {R}$ and all positive integers $n$ such that $\gcd(n,q)=1$. In some cases, the number of singular $n\times n$ circulant matrices over ${R}$ with a fixed determinant is determined through the link between the rings of circulant matrices and diagonal matrices. As applications, a brief discussion on the determinant of diagonal and circulant matrices over commutative finite principal ideal rings is given. Finally, some open problems and conjectures are posted