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Hon, Sean; Ayman-Mursaleen, Mohammad; Serra-Capizzano, Stefano

A note on the spectral distribution of symmetrized Toeplitz sequences. (English) Zbl 1419.15025

Linear Algebra Appl. 579, 32-50 (2019).
Summary: 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 N. L. Zamarashkin and E. E. Tyrtyshnikov [Sb. Math. 188, No. 8, 1191–1201 (1997; Zbl 0898.15007); translation from Mat. Sb. 188, No. 8, 83–92 (1997)], and P. Tilli [Linear Multilinear Algebra 45, No. 2–3, 147–159 (1998; Zbl 0951.65033)]. 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.

Cited in 9 Documents

MSC:

15B05 Toeplitz, Cauchy, and related matrices
65F15 Numerical computation of eigenvalues and eigenvectors of matrices
65F08 Preconditioners for iterative methods

Keywords:

Toeplitz matrices; Hankel matrices; circulant preconditioners

Citations:

Zbl 0898.15007; Zbl 0951.65033
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References:

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