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 $K_n(\rho )={\left[{\rho }^{|j-k|}\right]}_{j,k=1}^n$, where $\rho \in \mathbb{C}$. These are the level curves (contour lines) in the complex-$\rho$ plane on which $K_n(\rho )$ 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 $\rho ={\rho }_c$ at which multiple (double) eigenvalues occur. The present paper determines corresponding curves pertaining to eigenvalues of magnitude $N\ne n$. 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.