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Weyl distributions, spectral properties, and circulant approximation results for quaternion block multilevel Toeplitz matrix sequences

Ayoub Lailoune, Valerio Loi, Stefano Serra-Capizzano

TL;DR

The paper extends Weyl-type eigenvalue and singular value distribution theory to single-axis quaternion block multilevel Toeplitz sequences generated by quaternion-valued symbols. It builds a robust quaternion a.c.s. framework and shows that circulant matrix sequences provide effective approximations for preconditioning, all anchored by a symplectic embedding that transfers quaternionic problems to the complex GLT setting. Central contributions include precise canonical eigenvalue formulations, Schatten-norm estimates, and distribution results for both eigenvalues and singular values under left/right and sandwich kernels, along with practical circulant-a.c.s. constructions. Numerical experiments corroborate the theory across continuous and $L^{1}$ symbols, Hermitian and non-Hermitian cases, highlighting potential for fast, structure-preserving solvers in large-scale quaternion linear systems.

Abstract

The present work contains a comprehensive treatment of Weyl eigenvalue and singular value distributions for single-axis quaternion block multilevel Toeplitz matrix sequences generated by $s\times t$ quaternion matrix-valued, $d$-variate, Lebesgue integrable generating functions. Furthermore, in view of concrete applications, we are interested in preconditioning and matrix approximation results. To this end, a crucial step is the extension of the notion of an approximating class of sequences (a.c.s.) to the case of matrix sequences with quaternion entries, since it allows us to decompose the difference between a matrix and its preconditioner into low-norm plus (relatively) low-rank terms. As a specific example, we consider classes of quaternion block multilevel circulant matrix sequences as an a.c.s. for quaternion block multilevel Toeplitz matrix sequences. These approximation results lay the foundations for fast preconditioning methods when dealing with large quaternion linear systems stemming from modern applications. We conclude our study with numerical experiments and directions for future research.

Weyl distributions, spectral properties, and circulant approximation results for quaternion block multilevel Toeplitz matrix sequences

TL;DR

The paper extends Weyl-type eigenvalue and singular value distribution theory to single-axis quaternion block multilevel Toeplitz sequences generated by quaternion-valued symbols. It builds a robust quaternion a.c.s. framework and shows that circulant matrix sequences provide effective approximations for preconditioning, all anchored by a symplectic embedding that transfers quaternionic problems to the complex GLT setting. Central contributions include precise canonical eigenvalue formulations, Schatten-norm estimates, and distribution results for both eigenvalues and singular values under left/right and sandwich kernels, along with practical circulant-a.c.s. constructions. Numerical experiments corroborate the theory across continuous and symbols, Hermitian and non-Hermitian cases, highlighting potential for fast, structure-preserving solvers in large-scale quaternion linear systems.

Abstract

The present work contains a comprehensive treatment of Weyl eigenvalue and singular value distributions for single-axis quaternion block multilevel Toeplitz matrix sequences generated by quaternion matrix-valued, -variate, Lebesgue integrable generating functions. Furthermore, in view of concrete applications, we are interested in preconditioning and matrix approximation results. To this end, a crucial step is the extension of the notion of an approximating class of sequences (a.c.s.) to the case of matrix sequences with quaternion entries, since it allows us to decompose the difference between a matrix and its preconditioner into low-norm plus (relatively) low-rank terms. As a specific example, we consider classes of quaternion block multilevel circulant matrix sequences as an a.c.s. for quaternion block multilevel Toeplitz matrix sequences. These approximation results lay the foundations for fast preconditioning methods when dealing with large quaternion linear systems stemming from modern applications. We conclude our study with numerical experiments and directions for future research.

Paper Structure

This paper contains 29 sections, 34 theorems, 198 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Organization of the work.
  3. Definitions and preliminary tools
  4. Algebraic primitives
  5. Symplectic embedding
  6. Spectral tools for quaternion matrices
  7. Canonical eigenvalues.
  8. Function theory preliminaries and notation
  9. Left, right, and sandwich trigonometric polynomials
  10. Asymptotic spectral distribution of quaternion matrix sequences
  11. Definition of a.c.s.
  12. Single-axis quaternion Toeplitz sequences
  13. $d$-level quaternion Toeplitz matrices
  14. Linearity, Adjoint, and Hermitian properties
  15. Symplectic embedding of Toeplitz matrices and spectral/singular value distributions of Toeplitz matrix sequences
  16. ...and 14 more sections

Key Result

Lemma 2.1

$\Phi$ is an injective real-linear $*$-algebra map. In particular,

Figures (4)

  • Figure 1: $2\times2$ continuous Hermitian: spectral rearranged symbol (dark blue line) compared with rearranged eigenvalues (blue points). Size of levels: $n=2,8,32$; classes: (L/$S_{12}$/$S_{21}$/R).
  • Figure 2: $2\times2$ continuous non-Hermitian: singular value rearranged symbol (dark blue line) compared with rearranged singular values (blue points). Size of levels: $n=2,8,32$; classes: (L/$S_{12}$/$S_{21}$/R).
  • Figure 3: $L^1$$2\times2$ Hermitian: spectral rearranged symbol (dark blue line) compared with rearranged eigenvalues (blue points). Size of levels: $n=2,8,32$; classes: (L/$S_{12}$/$S_{21}$/R).
  • Figure 4: $L^1$$2\times2$ non-Hermitian: singular value rearranged symbol (dark blue line) compared with rearranged singular values (blue points). Size of levels: $n=2,8,32$; classes: (L/$S_{12}$/$S_{21}$/R).

Theorems & Definitions (76)

  • Lemma 2.1: $*$-algebra property
  • Definition 2.2: Right eigenpair
  • Lemma 2.3: Conjugacy classes
  • Lemma 2.4: zhang_quaternions_1997
  • Theorem 2.5: zhang_quaternions_1997
  • Theorem 2.6: Quaternion SVD, zhang_quaternions_1997
  • Definition 2.7: Schatten $p$-norms
  • Proposition 2.8: Symplectic embedding duplicates singular values
  • Corollary 2.9
  • Lemma 2.10
  • ...and 66 more