3-by-3 matrices with elliptical numerical range revisited

Abstract

According to Kippenhahn's classification, numerical ranges W(A) of unitarily irreducible 3x3 matrices A come in three possible shapes, an elliptical disk being one of them. The known criterion for the ellipticity of W(A) consists of several equations, involving the eigenvalues of A. It is shown herein that the set of 3x3 matrices satisfying these conditions is nowhere dense, i.e., one of the necessary conditions can be violated by an arbitratily small perturbation of the matrix and therefore by an insufficiently good numberical approximation of the eigenvalues. 

Moreover, necessary and sufficient conditions for a real A to have an elliptical W(A) are derived involving only the matrix coefficients and not requiring the knowledge of the eigenvalues. A particular case of real companion matrices is considered in detail.