Companion Matrices and Their Relations to Toeplitz and Hankel Matrices

TL;DR

This paper explores the properties of companion matrices and their connections to Toeplitz and Hankel matrices, revealing new conditions for similarity transformations and structure-preserving extensions.

Contribution

It introduces a generalized condition for Toeplitz and Hankel matrices to serve as similarity transformations between companion matrices, and shows how these matrices can be extended while preserving their structure.

Findings

New generalized condition for similarity transformations

Patterns in products of companion and Toeplitz/Hankel matrices

Extensions of Toeplitz/Hankel matrices preserving structure

Abstract

In this paper we describe some properties of companion matrices and demonstrate some special patterns that arise when a Toeplitz or a Hankel matrix is multiplied by a related companion matrix. We present a new condition, generalizing known results, for a Toeplitz or a Hankel matrix to be the transforming matrix for a similarity between a pair of companion matrices. A special case of our main result shows that a Toeplitz or a Hankel matrix can be extended using associated companion matrices, preserving the Toeplitz or Hankel structure respectively.