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

TL;DR

This paper explores the eigenvalue contour lines of Kac-Murdock-Szego matrices with complex parameters, revealing how these curves change shape and properties for eigenvalues of different magnitudes, and discusses their mathematical significance.

Contribution

It extends previous work by analyzing eigenvalue contour curves for eigenvalues of magnitude N≠n, showing they lack cusps and can form loops, and discusses their topological properties.

Findings

Curves for eigenvalues of magnitude N≠n lack cusps.

When N<n, curves form loops instead of cusps.

Winding numbers of the curves are discussed.

Abstract

A previous paper studied the so-called borderline curves of the Kac--Murdock--Szeg\H{o} 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.