Circular Hessenberg Pairs

TL;DR

This paper introduces and classifies recurrent circular Hessenberg pairs and systems, exploring their properties, bases, transition matrices, and relations, contributing to the understanding of special matrix pairs in linear algebra.

Contribution

It defines recurrent circular Hessenberg pairs and classifies their associated systems into four families, establishing their structure and relations.

Findings

Recurrent circular Hessenberg pairs satisfy tridiagonal relations.

Six bases are identified with explicit transition matrices.

Four families of recurrent circular Hessenberg systems are constructed.

Abstract

A square matrix is called Hessenberg whenever each entry below the subdiagonal is zero and each entry on the subdiagonal is nonzero. Let $M$ denote a Hessenberg matrix. Then $M$ is called circular whenever the upper-right corner entry of $M$ is nonzero and every other entry above the superdiagonal is zero. A circular Hessenberg pair consists of two diagonalizable linear maps on a nonzero finite-dimensional vector space, that each act on an eigenbasis of the other one in a circular Hessenberg fashion. Let $A, A^*$ denote a circular Hessenberg pair. We investigate six bases for the underlying vector space that we find attractive. We display the transition matrices between certain pairs of bases among the six. We also display the matrices that represent $A$ and $A^*$ with respect to the six bases. We introduce a special type of circular Hessenberg pair, said to be recurrent. We show that a circular Hessenberg pair $A, A^*$ is recurrent if and only if $A, A^*$ satisfy the tridiagonal relations. For a circular Hessenberg pair, there is a related object called a circular Hessenberg system. We classify up to isomorphism the recurrent circular Hessenberg systems. To this end, we construct four families of recurrent circular Hessenberg systems. We show that every recurrent circular Hessenberg system is isomorphic to a member of one of the four families.