Spectral properties of Kac-Murdock-Szeg\"o matrices with a complex parameter

When $0<\rho <1$, the Kac-Murdock-Szeg"o matrix $K_n(\rho )={\left[{\rho }^{\mid j-k\mid }\right]}_{j,k=1}^n$ is a Toeplitz correlation matrix with many applications and very well known spectral properties. We study the eigenvalues and eigenvectors of $K_n(\rho )$ for the general case where $\rho$ is complex, pointing out similarities and differences to the case $0<\rho <1$. We then specialize our results to real $\rho$ with $\rho >1$, emphasizing the continuity of the eigenvalues as functions of $\rho$. For $\rho >1$, we develop simple approximate formulas for the eigenvalues and pinpoint all eigenvalues’ locations. Our study starts from a certain polynomial whose zeros are connected to the eigenvalues by elementary formulas. We discuss relations of our results to earlier results of W. F. Trench.