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Normal form for GLT sequences, functions of normal GLT sequences, and spectral distribution of perturbed normal matrices

Giovanni Barbarino, Carlo Garoni

TL;DR

This paper extends the theory of generalized locally Toeplitz (GLT) sequences by proving that every GLT sequence admits a normal form and by identifying the spectral symbol for sequences formed by normal matrices. It further shows that if a GLT sequence is composed of normal matrices and a continuous function $f$ is applied to the matrices, then the resulting sequence is GLT with symbol $f(\kappa)$, linking functional calculus to GLT symbols. A spectral distribution result for perturbed normal matrices is established, demonstrating stability under small perturbations under precise norms or zero-distribution conditions. Collectively, these results provide a robust framework for predicting eigenvalue distributions of discretization matrices and for extending GLT methods to matrix functions and perturbations, with practical impact on spectral analysis in numerical PDE discretizations.

Abstract

The theory of generalized locally Toeplitz (GLT) sequences is a powerful apparatus for computing the asymptotic spectral distribution of matrices $A_n$ arising from numerical discretizations of differential equations. Indeed, when the mesh fineness parameter $n$ tends to infinity, these matrices $A_n$ give rise to a sequence $\{A_n\}_n$, which often turns out to be a GLT sequence. In this paper, we extend the theory of GLT sequences in several directions: we show that every GLT sequence enjoys a normal form, we identify the spectral symbol of every GLT sequence formed by normal matrices, and we prove that, for every GLT sequence $\{A_n\}_n$ formed by normal matrices and every continuous function $f:\mathbb C\to\mathbb C$, the sequence $\{f(A_n)\}_n$ is again a GLT sequence whose spectral symbol is $f(κ)$, where $κ$ is the spectral symbol of $\{A_n\}_n$. In addition, using the theory of GLT sequences, we prove a spectral distribution result for perturbed normal matrices.

Normal form for GLT sequences, functions of normal GLT sequences, and spectral distribution of perturbed normal matrices

TL;DR

This paper extends the theory of generalized locally Toeplitz (GLT) sequences by proving that every GLT sequence admits a normal form and by identifying the spectral symbol for sequences formed by normal matrices. It further shows that if a GLT sequence is composed of normal matrices and a continuous function is applied to the matrices, then the resulting sequence is GLT with symbol , linking functional calculus to GLT symbols. A spectral distribution result for perturbed normal matrices is established, demonstrating stability under small perturbations under precise norms or zero-distribution conditions. Collectively, these results provide a robust framework for predicting eigenvalue distributions of discretization matrices and for extending GLT methods to matrix functions and perturbations, with practical impact on spectral analysis in numerical PDE discretizations.

Abstract

The theory of generalized locally Toeplitz (GLT) sequences is a powerful apparatus for computing the asymptotic spectral distribution of matrices arising from numerical discretizations of differential equations. Indeed, when the mesh fineness parameter tends to infinity, these matrices give rise to a sequence , which often turns out to be a GLT sequence. In this paper, we extend the theory of GLT sequences in several directions: we show that every GLT sequence enjoys a normal form, we identify the spectral symbol of every GLT sequence formed by normal matrices, and we prove that, for every GLT sequence formed by normal matrices and every continuous function , the sequence is again a GLT sequence whose spectral symbol is , where is the spectral symbol of . In addition, using the theory of GLT sequences, we prove a spectral distribution result for perturbed normal matrices.

Paper Structure

This paper contains 26 sections, 18 theorems, 93 equations.

Table of Contents

  1. Introduction
  2. Overview of the theory of GLT sequences
  3. Singular value and spectral distribution of a sequence of matrices
  4. Special matrix-sequences
  5. Zero-distributed sequences
  6. Diagonal sampling sequences
  7. Toeplitz sequences
  8. Approximating classes of sequences
  9. GLT sequences
  10. Main results
  11. Proofs of the main results
  12. Proof of Theorem \ref{['normal']}
  13. Diagonal sequences
  14. Circulant sequences
  15. LT operator
  16. ...and 11 more sections

Key Result

Theorem 3.1

There exists a unitary sequence $\{Q_n\}_n$ with the following property. For every GLT sequence $\{A_n\}_n\sim_{\rm GLT}\kappa$ there exists a diagonal sequence $\{D_n\}_n$ such that $\{A_n\}_n\equiv_{\rm a.c.s.}\{Q_nD_nQ_n^*\}_n$, $\{D_n\}_n\sim_\lambda\kappa$, and $D_n$ is real for every $n$ if $\

Theorems & Definitions (45)

  • Definition 2.1: singular value and spectral distribution of a sequence of matrices
  • Remark 2.1
  • Definition 2.2: approximating class of sequences
  • Definition 2.3: GLT sequence
  • Theorem 3.1: normal form for GLT sequences
  • Theorem 3.2: spectral distribution of perturbed normal sequences
  • Theorem 3.3: spectral distribution of normal GLT sequences
  • Theorem 3.4: functions of normal GLT sequences
  • Lemma 4.1
  • proof
  • ...and 35 more