GLT SEQUENCES AND NORMAL MATRICES

Generalized locally Toeplitz sequences; Matrix functions; Normal form; Normal matrices and perturbed normal matrices; Spectral distribution; Algebra and Number Theory

Abstract :

[en] The theory of generalized locally Toeplitz (GLT) sequences is an apparatus for computing the asymptotic spectral distribution of matrices An arising from numerical discretizations of differential equations. Indeed, when the mesh fineness parameter n tends to infinity, these matrices An give rise to a sequence {An}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 {An}n formed by normal matrices and every continuous function f: C → C, the sequence {f(An)}n is again a GLT sequence whose spectral symbol is f(κ), where κ is the spectral symbol of {An}n. In addition, using the theory of GLT sequences, we prove a spectral distribution result for perturbed normal matrices.

Disciplines :

Mathematics

Author, co-author :

Barbarino, Giovanni ; Université de Mons - UMONS > Recherche > Service ERC Unit - Matrix Theory and Optimization

Garoni, Carlo ; Department of Mathematics, University of Rome Tor Vergata, Italy

Language :

English

Title :

GLT SEQUENCES AND NORMAL MATRICES

Publication date :

2025

Journal title :

Electronic Journal of Linear Algebra

ISSN :

1537-9582

eISSN :

1081-3810

Publisher :

International Linear Algebra Society

Volume :

Pages :

1 - 20

Peer reviewed :

Peer Reviewed verified by ORBi

Additional URL :

https://journals.uwyo.edu/index.php/ela/article/download/8929/6949

Research unit :

F151 - Mathématique et Recherche opérationnelle

Research institute :

R300 - Institut de Recherche en Technologies de l'Information et Sciences de l'Informatique
R450 - Institut NUMEDIART pour les Technologies des Arts Numériques

Funding text :

The authors are members of the Research Group GNCS (Gruppo Nazionale per il Calcolo Scientifico) of INdAM (Istituto Nazionale di Alta Matematica). Giovanni Barbarino was supported by an Academy of Finland grant (Suomen Akatemian P\u00E4\u00E4t\u00F6s 331240), by the Alfred Kordelinin S\u00E4\u00E4ti\u00F6 Grant 210122, and by the ERC Consolidator Grant 101085607 through the Project eLinoR. Carlo Garoni was supported by the MUR Excellence Department Projects Math@TOV (CUP E83C18000100006) and MatMod@TOV (CUP E83C23000330006) awarded to the Department of Mathematics of the University of Rome Tor Vergata, by the Department of Mathematics of the University of Rome Tor Vergata through the Project RICH GLT (CUP E83C22001650005), and by an INdAM-GNCS Project (CUP E53C22001930001).Acknowledgments. The authors are members of the Research Group GNCS (Gruppo Nazionale per il Calcolo Scientifico) of INdAM (Istituto Nazionale di Alta Matematica). Giovanni Barbarino was supported by an Academy of Finland grant (Suomen Akatemian P\u00E4\u00E4t\u00F6s 331240), by the Alfred Kordelinin S\u00E4\u00E4ti\u00F6 Grant 210122, and by the ERC Consolidator Grant 101085607 through the Project eLinoR. Carlo Garoni was supported by the MUR Excellence Department Projects Math@TOV (CUP E83C18000100006) and MatMod@TOV (CUP E83C23000330006) awarded to the Department of Mathematics of the University of Rome Tor Vergata, by the Department of Mathematics of the University of Rome Tor Vergata through the Project RICH GLT (CUP E83C22001650005), and by an INdAM-GNCS Project (CUP E53C22001930001).

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since 21 December 2025

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