Double, borderline, and extraordinary eigenvalues of Kac-Murdock-Szeg\"o matrices with a complex parameter

For all sufficiently large complex $\rho$, and for arbitrary matrix dimension $n$, it is shown that the Kac--Murdock--Szeg\H{o} matrix $K_n(\rho)=\left[\rho^{|j-k|}\right]_{j,k=1}^{n}$ possesses exactly two eigenvalues whose magnitude is larger than $n$. We discuss a number of properties of the two "extraordinary" eigenvalues. Conditions are developed that, given $n$, allow us-without actually computing eigenvalues-to find all values $\rho$ that give rise to eigenvalues of magnitude $n$, termed "borderline" eigenvalues. The aforementioned values of $\rho$ form two closed curves in the complex-$\rho$ plane. We describe these curves, which are $n$-dependent, in detail. An interesting borderline case arises when an eigenvalue of $K_n(\rho)$ equals $-n$: apart from certain exceptional cases, this occurs if and only if the eigenvalue is a double one; and if and only if the point $\rho$ is a cusp-like singularity of one of the two closed curves.