Eigenvalue bifurcations in Kac-Murdock-Szego matrices with a complex parameter

TL;DR

This paper investigates how eigenvalues of Kac-Murdock-Szegő matrices change as the complex parameter varies, focusing on bifurcations and their relation to borderline curves in the complex plane.

Contribution

It provides a detailed analysis of eigenvalue bifurcations in Kac-Murdock-Szegő matrices with complex parameters, linking bifurcation phenomena to borderline curves.

Findings

Eigenvalue bifurcations occur at specific parameter values.

Bifurcations are connected to the geometry of borderline curves.

Quantitative descriptions of bifurcation points are provided.

Abstract

For complex $\rho$, the spectral properties of the Toeplitz matrix $K_{n}(\rho)=\left[\rho^{|j-k|}\right]_{j,k=1}^{n}$, often called the Kac-Murdock-Szeg{\omicron} matrix, have been examined in detail in two recent papers. The second paper, in particular, introduced the concept of borderline curves. These are two closed curves in the complex-$\rho$ plane that consist of all the $\rho$ for which $K_n(\rho)$ possesses some eigenvalue whose magnitude equals the matrix dimension $n$. 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.