3-by-3 matrices with elliptical numerical range revisited

According to Kippenhahn’s classification, numerical ranges W (A) of unitarily irreducible 3 × 3 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 3× 3 matrices satisfying these conditions is nowhere dense, i.e., one of the necessary conditions can be violated by an arbitrarily small perturbation of the matrix, and therefore by an insufficiently good numerical 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.