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

TL;DR

This paper advances the theory of GLT sequences by establishing a normal form, analyzing spectral symbols of normal matrices, and demonstrating that functions of such matrices preserve the GLT structure, with applications to spectral distribution of perturbed normal matrices.

Contribution

It introduces a normal form for GLT sequences, characterizes spectral symbols of normal GLT sequences, and shows that applying functions to these sequences preserves their GLT structure.

Findings

Every GLT sequence has a normal form.

Spectral symbols of normal GLT sequences are identified.

Functions of normal GLT sequences produce new GLT sequences with spectral symbol f(κ).

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 $f(\kappa)$, where $\kappa$ 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.