Thin Hessenberg Pairs

TL;DR

This paper introduces and classifies thin Hessenberg pairs and systems, exploring their matrix representations and basis transformations, contributing to the understanding of their algebraic structure.

Contribution

It defines thin Hessenberg pairs and systems, provides matrix and basis characterizations, and classifies these systems up to isomorphism.

Findings

Characterization of TH pairs via basis transformations

Explicit matrix forms and transition matrices

Classification of TH systems up to isomorphism

Abstract

A square matrix is called {\it Hessenberg} whenever each entry below the subdiagonal is zero and each entry on the subdiagonal is nonzero. Let $V$ denote a nonzero finite-dimensional vector space over a field $\fld$. We consider an ordered pair of linear transformations $A: V \to V$ and $A^*: V \to V$ which satisfy both (i), (ii) below. \begin{enumerate} \item There exists a basis for $V$ with respect to which the matrix representing $A$ is Hessenberg and the matrix representing $A^*$ is diagonal. \item There exists a basis for $V$ with respect to which the matrix representing $A$ is diagonal and the matrix representing $A^*$ is Hessenberg. \end{enumerate} \noindent We call such a pair a {\it thin Hessenberg pair} (or {\it TH pair}). This is a special case of a {\it Hessenberg pair} which was introduced by the author in an earlier paper. We investigate several bases for $V$ with respect to which the matrices representing $A$ and $A^*$ are attractive. We display these matrices along with the transition matrices relating the bases. We introduce an "oriented" version of $A,A^*$ called a TH system. We classify the TH systems up to isomorphism.