Block structured matrix-sequences and their spectral and singular value canonical distributions: a general theory

TL;DR

This paper develops a general theoretical framework for analyzing the spectral and singular value distributions of block-structured matrix sequences, extending existing theories to more complex block configurations.

Contribution

It introduces a comprehensive theory for block matrix sequences with unilevel Toeplitz blocks, addressing gaps in the asymptotic distribution analysis for complex structures.

Findings

The theory applies to matrices modeling real-world infinite-dimensional operators.

Numerical tests validate the theoretical predictions.

Open problems in the field are discussed.

Abstract

In recent years more and more involved block structures appeared in the literature in the context of numerical approximations of complex infinite dimensional operators modeling real-world applications. In various settings, thanks the theory of generalized locally Toeplitz matrix-sequences, the asymptotic distributional analysis is well understood, but a general theory is missing when general block structures are involved. The central part of the current work deals with such a delicate generalization when blocks are of (block) unilevel Toeplitz type, starting from a problem of recovery with missing data. Visualizations, numerical tests, and few open problems are presented and critically discussed.