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Structural properties of a symmetric Toeplitz and Hankel matrices

Hojin Chu, Homoon Ryu

TL;DR

This work introduces weighted Toeplitz and Hankel graphs to study symmetric Toeplitz and Hankel matrices through their graph components. It proves that the Frobenius normal form decomposes into a direct sum of irreducible blocks: for Toeplitz matrices these blocks form a chain under principal-submatrix inclusion, while for Hankel matrices the blocks are irreducible but need not nest. Components of the associated graphs determine the diagonal blocks, enabling a graph-based route to compute the Frobenius form and highlighting a key structural distinction between Toeplitz and Hankel matrices. The results connect graph theory with canonical matrix decompositions and provide a constructive framework for identifying irreducible blocks from the graph components.

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.

Structural properties of a symmetric Toeplitz and Hankel matrices

TL;DR

This work introduces weighted Toeplitz and Hankel graphs to study symmetric Toeplitz and Hankel matrices through their graph components. It proves that the Frobenius normal form decomposes into a direct sum of irreducible blocks: for Toeplitz matrices these blocks form a chain under principal-submatrix inclusion, while for Hankel matrices the blocks are irreducible but need not nest. Components of the associated graphs determine the diagonal blocks, enabling a graph-based route to compute the Frobenius form and highlighting a key structural distinction between Toeplitz and Hankel matrices. The results connect graph theory with canonical matrix decompositions and provide a constructive framework for identifying irreducible blocks from the graph components.

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.

Paper Structure

This paper contains 5 sections, 9 theorems, 48 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The Frobenius normal form of a symmetric Toeplitz matrix
  4. The Frobenius normal form of a Hankel matrix
  5. Acknowledgement

Key Result

Theorem 2.1

Let $A$ be a square matrix of order $n$. Then $A$ is irreducible if and only if the (weighted) graph having adjacency matrix $A$ is connected.

Figures (2)

  • Figure 1: The symmetric Toeplitz matrix $A= T[0,0,3,0,8,0,9]$ and its graph
  • Figure 2: The Hankel matrix $A= H[0,3,0,5,0,0,0,0,0,0,0,2,2,4,0]$ and its graph

Theorems & Definitions (19)

  • Definition 1.1
  • Definition 1.2
  • Theorem 2.1
  • Lemma 3.1
  • proof
  • Theorem 3.2
  • proof
  • Lemma 3.3
  • Theorem 3.4
  • proof
  • ...and 9 more