Unified approach to reciprocal matrices with Kippenhahn curves containing elliptical components

We consider tridiagonal matrices $ (a_{ij})_{i,j=1}<^>n $ (aij)i,j=1n with constant main diagonal and such that $ a_{i,i+1}a_{i+1,i}=1 $ ai,i+1ai+1,i=1 for $ i=1,\ldots,n-1 $ i=1,& mldr;,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 a system of homogeneous polynomial equations in variables $ (\vert {a_{j,j+1}}\vert -\vert {a_{j+1,j}}\vert )<^>2 $ (|aj,j+1|-|aj+1,j|)2, and established via a unified approach across arbitrary dimensions. The results are illustrated, and specific numerical examples are provided for n = 7, thus generalizing earlier work in the lower-dimensional setting.