Unified approach to reciprocal matrices with Kippenhahn curves containing elliptical components

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.