Eigenvalue contour lines of Kac-Murdock-Szego matrices with a complex parameter

Abstract

A previous paper studied the so-called borderline curves of the Kac-Murdock-Szegő matrix Kn(ρ) = ρ |j−k| n j,k=1 , where ρ ∈ C. These are the level curves (contour lines) in the complex-ρ plane on which Kn(ρ) has a type-1 or type-2 eigenvalue of magnitude n, where n is the matrix dimension. Those curves have cusps at all critical points ρ = ρc at which multiple (double) eigenvalues occur. The present paper determines corresponding curves pertaining to eigenvalues of magnitude N = n. We find that these curves no longer present cusps; and that, when N < n, the cusps have in a sense transformed into loops. We discuss the meaning of the winding numbers of our curves. Finally, we point out possible extensions to more general matrices.