Low-dimensional reciprocal matrices with elliptical components of their Kippenhahn curves

TL;DR

This paper characterizes when the Kippenhahn curves of low-dimensional reciprocal matrices are composed solely of elliptical components, providing criteria and a complete description of higher-rank numerical ranges in such cases.

Contribution

It establishes criteria for elliptical components in Kippenhahn curves of reciprocal matrices up to size six and describes higher-rank numerical ranges under these conditions.

Findings

Criteria for elliptical components in Kippenhahn curves for n ≤ 6

Complete description of higher-rank numerical ranges when criteria are met

Characterization of low-dimensional reciprocal matrices' numerical ranges

Abstract

By definition, reciprocal matrices are tridiagonal $n$-by-$n$ matrices $A$ with constant main diagonal and such that $a_{i,i+1}a_{i+1,i}=1$ for $i=1,\ldots,n-1$. For $n\leq 6$, we establish criteria under which the numerical range generating curves (also called Kippenhahn curves) of such matrices consist of elliptical components only. As a corollary, we also provide a complete description of higher-rank numerical ranges when the criteria are met.