Some (Hopf) algebraic properties of circulant matrices

TL;DR

This paper explores the algebraic structures of circulant matrices, revealing their connection to group algebras, and introduces a new class of matrices with easily accessible eigenvalues and eigenvectors.

Contribution

It provides a detailed analysis of the Hopf algebraic properties of circulant matrices and introduces a generalized matrix class with explicit spectral characteristics.

Findings

Circulant matrices are isomorphic to group algebras of cyclic groups.

A new class of matrices generalizing circulant and skew circulant matrices is proposed.

Eigenvalues and eigenvectors of the new matrices can be directly obtained from their entries.

Abstract

We study some (Hopf) algebraic properties of circulant matrices, inspired by the fact that the algebra of circulant $n\times n$ matrices is isomorphic to the group algebra of the cyclic group with $n$ elements. We introduce also a class of matrices that generalize both circulant and skew circulant matrices, and for which the eigenvalues and eigenvectors can be read directly from their entries.