It is known that the generating function f of a sequence of Toeplitz matrices {Tn(f)}n may not describe the asymptotic distribution of the eigenvalues of Tn(f) if f is not real. In a recent paper, we assume as a working hypothesis that, if the eigenvalues of Tn(f) are real for all n, then they admit an asymptotic expansion where the first function g appearing in this expansion is real and describes the asymptotic distribution of the eigenvalues of Tn(f). In this paper we extend this idea to Toeplitz matrices with complex eigenvalues. The paper is predominantly a numerical exploration of different typical cases, and presents several avenues of possible future research.
@article{arxiv.1910.13810,
title = {A Matrix-Less Method to Approximate the Spectrum and the Spectral Function of Toeplitz Matrices with Complex Eigenvalues},
author = {Sven-Erik Ekström and Paris Vassalos},
journal= {arXiv preprint arXiv:1910.13810},
year = {2019}
}