A note on the spectral distribution of symmetrized Toeplitz sequences

Sean Y S HON*, Mohammad Ayman Mursaleen, Stefano Serra-Capizzano

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

4 Citations (Scopus)

Abstract

The singular value and spectral distribution of Toeplitz matrix sequences with Lebesgue integrable generating functions is well studied. Early results were provided in the classical Szegő theorem and the Avram-Parter theorem, in which the singular value symbol coincides with the generating function. More general versions of the theorem were later proved by Zamarashkin and Tyrtyshnikov, and Tilli. Considering (real)nonsymmetric Toeplitz matrix sequences, we first symmetrize them via a simple permutation matrix and then we show that the singular value and spectral distribution of the symmetrized matrix sequence can be obtained analytically, by using the notion of approximating class of sequences. In particular, under the assumption that the symbol is sparsely vanishing, we show that roughly half of the eigenvalues of the symmetrized Toeplitz matrix (i.e. a Hankel matrix)are negative/positive for sufficiently large dimension, i.e. the matrix sequence is symmetric (asymptotically)indefinite.

Original languageEnglish
Pages (from-to)32-50
Number of pages19
JournalLinear Algebra and Its Applications
Volume579
DOIs
Publication statusPublished - 15 Oct 2019

Scopus Subject Areas

  • Algebra and Number Theory
  • Numerical Analysis
  • Geometry and Topology
  • Discrete Mathematics and Combinatorics

User-Defined Keywords

  • Circulant preconditioners
  • Hankel matrices
  • Toeplitz matrices

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