Unified approach to reciprocal matrices with Kippenhahn curves containing elliptical components

TL;DR

This paper develops a unified method to determine when reciprocal matrices have Kippenhahn curves with elliptical components, generalizing previous lower-dimensional results through polynomial criteria applicable to any size.

Contribution

It introduces a unified approach using polynomial equations to analyze Kippenhahn curves of reciprocal matrices across all dimensions, extending earlier lower-dimensional findings.

Findings

Criteria for elliptical components in Kippenhahn curves are established.

The approach applies to matrices of arbitrary size, demonstrated with n=7.

Numerical examples illustrate the theoretical results.

Abstract

Reciprocal matrices are tridiagonal matrices $(a_{ij})_{i,j=1}^n$ with constant main diagonal and such that $a_{i,i+1}a_{i+1,i}=1$ for $i=1,\ldots,n-1$. For these matrices, criteria are established under which their Kippenhahn curves contain elliptical components or even consist completely of such. These criteria are in terms of system of homogeneous polynomial equations in variables $(\left|a_{j,j+1}\right|-\left|a_{j+1,j}\right|)^2$, and established via a unified approach across arbitrary dimensions. The results are illustrated, and specific numerical examples provided, for $n=7$ thus generalizing earlier work in the lower dimensional setting.