The numerical range of \(3 \times 3\) matrices (Q676023)

Let \(A\) be an \(n\times n\) complex matrix and \(W(A)=\{x^*Ax:x\in \mathbb{C}^n\), \(x^*x=1\}\) the numerical range of \(A\). The main purpose of this paper is to establish a series of tests allowing one to determine the shape of \(W(A)\) for \(n=3\). The authors also examine the reconstruction of \(A\) from \(W(A)\) for \(n=3\) up to unitary similarity.