Structural properties of a symmetric Toeplitz and Hankel matrices

TL;DR

This paper explores the structural properties of symmetric Toeplitz and Hankel matrices by analyzing their associated weighted graphs, revealing how their Frobenius normal forms decompose into irreducible components.

Contribution

It introduces the concepts of weighted Toeplitz and Hankel graphs and characterizes the Frobenius normal forms of these matrices through graph component analysis.

Findings

Frobenius normal form of symmetric Toeplitz matrices is a direct sum of symmetric irreducible Toeplitz matrices.

Frobenius normal form of Hankel matrices is a direct sum of irreducible Hankel matrices.

Graph component analysis provides insights into the matrix decompositions.

Abstract

In this paper, we investigate properties of a symmetric Toeplitz matrix and a Hankel matrix by studying the components of its graph. To this end, we introduce the notion of ``weighted Toeplitz graph" and ``weighted Hankel graph", which are weighted graphs whose adjacency matrix are a symmetric Toeplitz matrix and a Hankel matrix, respectively. By studying the components of a weighted Toeplitz graph, we show that the Frobenius normal form of a symmetric Toeplitz matrix is a direct sum of symmetric irreducible Toeplitz matrices. Similarly, by studying the components of a weighted Hankel matrix, we show that the Frobenius normal form of a Hankel matrix is a direct sum of irreducible Hankel matrices.