Elliptic numerical ranges of 4 × 4 matrices

Abstract

Let A be an n × n (complex) matrix. Recall that the numerical range W(A) of A is the set {Ax, x : x ∈ Cn, x = 1} in the plane, where ·, · denotes the usual inner product in Cn. In this paper a series of tests is given, allowing one to determine when the numerical range of a 4 × 4 matrix A is an elliptic disc.