The Eigenvalue Distribution of Special 2-by-2 Block Matrix-Sequences with Applications to the Case of Symmetrized Toeplitz Structures

Paola Ferrari, Isabella Furci, Sean Hon, Mohammad Ayman Mursaleen, Stefano Serra-Capizzano

Research output: Contribution to journalJournal articlepeer-review

23 Citations (Scopus)
22 Downloads (Pure)

Abstract

Given a Lebesgue integrable function f over [−π, π], we consider the sequence of matrices {YnTn[f]}n, where Tn[f] is the n-by-n Toeplitz matrix generated by f and Yn is the anti-identity matrix. Because of the unitary nature of Yn, the singular values of Tn[f] and YnTn[f] coincide. However, the eigenvalues are affected substantially by the action of Yn. Under the assumption that the Fourier coefficients of f are real, we prove that {YnTn[f]}n is distributed in the eigenvalue sense as ±|f|. A generalization of this result to the block Toeplitz case is also shown. We also consider the preconditioning introduced by [J. Pestana and A. Wathen, SIAM J. Matrix Anal. Appl., 36 (2015), pp. 273–288] and prove that the preconditioned matrix-sequence is distributed in the eigenvalue sense as φ1 under the mild assumption that f is sparsely vanishing. We emphasize that the mathematical tools introduced in this setting have a general character and can be potentially used in different contexts. A number of numerical experiments are provided and critically discussed.

Original languageEnglish
Pages (from-to)1066-1086
Number of pages21
JournalSIAM Journal on Matrix Analysis and Applications
Volume40
Issue number3
DOIs
Publication statusPublished - 12 Sept 2019

Scopus Subject Areas

  • Analysis

User-Defined Keywords

  • Circulant preconditioners
  • Eigenvalue distribution
  • Hankel matrices
  • Singular value distribution
  • Toeplitz matrices

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