  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.
