A note on the spectral distribution of symmetrized Toeplitz sequences

TL;DR

This paper investigates the spectral distribution of symmetrized Toeplitz matrix sequences, showing that under certain conditions, their eigenvalues are asymptotically split between positive and negative, revealing indefinite behavior.

Contribution

It introduces a method to analyze the spectral distribution of symmetrized Toeplitz matrices using approximation classes, extending classical results to nonsymmetric cases.

Findings

Eigenvalues of symmetrized Toeplitz matrices are asymptotically split between positive and negative.

Under sparsely vanishing symbols, the matrices are asymptotically indefinite.

The spectral distribution can be analytically obtained using approximation class concepts.

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 Szeg{\H{o}} 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.