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

Abstract

When 0<ρ<1, the Kac–Murdock–Szegö matrix Kn(ρ)=[ρ|j−k|]j,k=1n is a Toeplitz correlation matrix with many applications and very well known spectral properties. We study the eigenvalues and eigenvectors of Kn(ρ) for the general case where ρ is complex, pointing out similarities and differences to the case 0<ρ<1. We then specialize our results to real ρ with ρ>1, emphasizing the continuity of the eigenvalues as functions of ρ. For ρ>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.