Abstract
We consider 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 , … , 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 ( | a j , j + 1 | − | a j + 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.
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