TY - JOUR

T1 - The Eigenvalue Distribution of Special 2-by-2 Block Matrix-Sequences with Applications to the Case of Symmetrized Toeplitz Structures

AU - Ferrari, Paola

AU - Furci, Isabella

AU - Hon, Sean

AU - Mursaleen, Mohammad Ayman

AU - Serra-Capizzano, Stefano

N1 - Funding Information:
The work of the first, second, and fifth authors was partially supported by the INdAM Research Group GNCS GNCS2018 project “Tecniche innovative per problemi di algebra lineare.”
Publisher copyright:
© 2019, Society for Industrial and Applied Mathematics

PY - 2019/9/12

Y1 - 2019/9/12

N2 - 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.

AB - 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.

KW - Circulant preconditioners

KW - Eigenvalue distribution

KW - Hankel matrices

KW - Singular value distribution

KW - Toeplitz matrices

UR - http://www.scopus.com/inward/record.url?scp=85072961313&partnerID=8YFLogxK

U2 - 10.1137/18M1207399

DO - 10.1137/18M1207399

M3 - Article

AN - SCOPUS:85072961313

SN - 0895-4798

VL - 40

SP - 1066

EP - 1086

JO - SIAM Journal on Matrix Analysis and Applications

JF - SIAM Journal on Matrix Analysis and Applications

IS - 3

ER -