Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A note on the spectral distribution of non-Hermitian block matrices with Toeplitz blocks

Andrea Adriani, Giacomo Tento

Abstract

In the present paper, we are concerned with the study of the spectral distribution of matrix-sequences showing a non-Hermitian block structure with Toeplitz blocks. We use the notion of geometric mean of matrices and the theory of Generalized Locally Toeplitz (GLT) sequences to perform our analysis and produce some numerical tests and visualizations to confirm our theoretical derivations.

A note on the spectral distribution of non-Hermitian block matrices with Toeplitz blocks

Abstract

In the present paper, we are concerned with the study of the spectral distribution of matrix-sequences showing a non-Hermitian block structure with Toeplitz blocks. We use the notion of geometric mean of matrices and the theory of Generalized Locally Toeplitz (GLT) sequences to perform our analysis and produce some numerical tests and visualizations to confirm our theoretical derivations.

Paper Structure

This paper contains 11 sections, 12 theorems, 66 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminary tools: Spectral distribution, Toeplitz matrices, GLT sequences
  3. Toeplitz matrices, Diagonal samplings and zero distributed sequences
  4. GLT sequences and Geometric means of matrices
  5. Proof of the main result
  6. Numerical tests
  7. Case $k=2$ (the easy setting)
  8. Case $k=2$ (the limit case)
  9. Case $k=3$ (the limit case)
  10. Case $k=3$ without technical assumptions
  11. Conclusions

Key Result

Theorem 1.1

Let $\mathcal{A}_n$ be as in Equation eq:A_Toeplitz. Assume that $f_{i,j} \in L^{\infty}([-\pi,\pi])$ and real-valued almost everywhere for every $i,j$. Moreover, assume that $\mathop{\mathrm{ess\,inf}} f_{i, i+1},\mathop{\mathrm{ess\,inf}} f_{i+1, i} > 0$ for every $i=1,\dots,k-1$. Then with Equivalently, with $\blacktriangleleft$$\blacktriangleleft$

Figures (5)

  • Figure 4.1: Comparison between the real part of the spectrum of $\mathcal{A}_n$ and a uniform sampling of $F(\theta)$ for $n=100,200$
  • Figure 4.2: Comparison between the real part of the spectrum of $\mathcal{A}_n$ and a uniform sampling of $F(\theta)$ for $n=100,200$
  • Figure 4.3: Comparison between the real part of the spectrum of $\mathcal{A}_n$ and a uniform sampling of $F(\theta)$ for $n=100,200$
  • Figure 4.4: Comparison between the real part of the spectrum of $\mathcal{A}_n$ and a uniform sampling of $F(\theta)$ for $n=100,200$
  • Figure 4.5: In each column, depending on $n$, it is represented the value of the imaginary part of all the eigenvalues of $\mathcal{A}_n$

Theorems & Definitions (22)

  • Theorem 1.1
  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Theorem 2.6
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • ...and 12 more