On reality of eigenvalues of banded block Toeplitz matrices

TL;DR

This paper proposes a conjecture characterizing when the eigenvalues of real banded block Toeplitz matrices are real, supported by numerical experiments and generalizing previous conjectures.

Contribution

It formulates and partially proves a general conjecture on the eigenvalue reality of banded block Toeplitz matrices, extending existing results.

Findings

Numerical experiments support the conjecture.

The conjecture generalizes previous results for banded Toeplitz matrices.

Necessary and sufficient conditions for eigenvalue reality are provided.

Abstract

We formulate and partially prove a general conjecture providing necessary and sufficient conditions for the reality of the asymptotic spectrum of an arbitrary real banded block Toeplitz matrix. Additionally we present numerical experiments supporting it. This conjecture is a direct generalization of the already existing one in the case of banded Toeplitz matrices.