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

For complex $ρ$, the spectral properties of the Toeplitz matrix $K_{n}(ρ)=\left[ρ^{|j-k|}\right]_{j,k=1}^{n}$, often called the Kac-Murdock-Szegο 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-$ρ$ plane that consist of all the $ρ$ for which $K_n(ρ)$ 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.