Spectral properties of Kac-Murdock-Szegö matrices with a complex parameter

Abstract

When $0\lt ρ\lt 1$, the Kac-Murdock-Szegö matrix $K_n(ρ)=\left[ρ^{\lvert j-k \rvert}\right]_{j,k=1}^n$ is a Toeplitz correlation matrix with many applications and very well known spectral properties. We study the eigenvalues and eigenvectors of $K_n(ρ)$ for the general case where $ρ$ is complex, pointing out similarities and differences to the case $0\lt ρ\lt 1$. We then specialize our results to real $ρ$ with $ρ\gt 1$, emphasizing the continuity of the eigenvalues as functions of $ρ$. For $ρ\gt 1$, 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.