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

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