Eigenvalue bifurcations in Kac-Murdock-Szegő matrices with a complex parameter (Q2226402)

Let \(\rho\) be a complex number. We define the Toeplitz matrix \(Kn(\rho)=[\rho^{|j-k|}]_{j,k=1}^n\) (often called the Kac-Murdock-Szegö matrix). This kind of matrices has been examined in detail in [\textit{G. Fikioris}, Linear Algebra Appl. 553, 182--210 (2018; Zbl 1391.15095); \textit{G. Fikioris} and \textit{T. K. Mavrogordatos}, Linear Algebra Appl. 575, 314--333 (2019; Zbl 1414.15010)]. The second paper, in particular, introduced the concept of borderline curves. These are two closed curves in the complex plane that consist of all the \(\rho\) for which \(K_n(\rho)\) has some eigenvalue whose magnitude equals the matrix dimension. The purpose of the present paper is to examine eigenvalue bifurcations in both a qualitative and a quantitative manner and to discuss connections between bifurcations and the borderline curves.