Eigenvalue contour lines of Kac–Murdock–Szegő matrices with a complex parameter

A previous paper studied the so-called borderline curves of the Kac–Murdock–Szegő matrix Kn(ρ)=[ρ|j−k|]j,k=1n, 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 modulus 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 modulus ν≠n. We find that these curves no longer present cusps; and that, when ν