ELLIPTIC NUMERICAL RANGES OF $4 \times 4$ MATRICES

Abstract

Let $A$ be an $n \times n$ (complex) matrix. Recall that the numerical range $W(A)$ of $A$ is the set $\{ \langle Ax,x \rangle : x \in \mathbb{C}^n, \|x\| = 1 \}$ in the plane, where $\langle \cdot,\cdot \rangle$ denotes the usual inner product in $\mathbb{C}^n$. In this paper a series of tests is given, allowing one to determine when the numerical range of a $4 \times 4$ matrix $A$ is an elliptic disc.