A Matrix-Less Method to Approximate the Spectrum and the Spectral Function of Toeplitz Matrices with Complex Eigenvalues

10/30/2019

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by   Sven-Erik Ekström, et al.

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It is known that the generating function f of a sequence of Toeplitz matrices {T_n(f)}_n may not describe the asymptotic distribution of the eigenvalues of T_n(f) if f is not real. In a recent paper, we assume as a working hypothesis that, if the eigenvalues of T_n(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 T_n(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.