TY - JOUR

T1 - A note on the spectral distribution of symmetrized Toeplitz sequences

AU - HON, Sean Y S

AU - Mursaleen, Mohammad Ayman

AU - Serra-Capizzano, Stefano

N1 - Funding Information:
Sean Hon acknowledges that his work was partially finished at the Numerical Analysis Group of Mathematical Institute, University of Oxford and thanks Andy Wathen for the fruitful discussion. The research of Stefano Serra-Capizzano is partially supported by INdAM GNCS (Gruppo Nazionale per il Calcolo Scientifico).

PY - 2019/10/15

Y1 - 2019/10/15

N2 - 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.

AB - 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.

KW - Circulant preconditioners

KW - Hankel matrices

KW - Toeplitz matrices

UR - http://www.scopus.com/inward/record.url?scp=85066291450&partnerID=8YFLogxK

U2 - 10.1016/j.laa.2019.05.027

DO - 10.1016/j.laa.2019.05.027

M3 - Journal article

AN - SCOPUS:85066291450

SN - 0024-3795

VL - 579

SP - 32

EP - 50

JO - Linear Algebra and Its Applications

JF - Linear Algebra and Its Applications

ER -