Double, borderline, and extraordinary eigenvalues of Kac–Murdock–Szegő matrices with a complex parameter

Abstract

For all sufficiently large complex ρ, and for arbitrary matrix dimension n, it is shown that the Kac–Murdock–Szegő matrix Kn(ρ)=[ρ|j−k|]j,k=1n 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 ρ that give rise to eigenvalues of magnitude n, termed “borderline” eigenvalues. The aforementioned values of ρ form two closed curves in the complex-ρ plane. We describe these curves, which are n-dependent, in detail. An interesting borderline case arises when an eigenvalue of Kn(ρ) 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 ρ is a cusp-like singularity of one of the two closed curves.