Elliptic numerical ranges of 4 × 4 matrices

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.