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

Abstract

The present work contains a comprehensive treatment of Weyl eigenvalue and singular value distributions, \val{Schatten $p$-norm estimates, spectral localization and positive-definiteness criteria} for single-axis quaternion block multilevel Toeplitz matrix sequences generated by $s\times t$ quaternion matrix-valued, $d$-variate, Lebesgue integrable generating functions\val{, where $s,t,d$ are positive integers and where spectral localization and positive-definiteness are studied in the Hermitian setting, with $s=t$}. 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.