Numerical range of a doubly stochastic matrix (Q807719)

The numerical range of an \(n\times n\) complex matrix A is the set \(W(A)=\{x^*Ax:\) \(x\in {\mathbb{C}}^ n\), \(x^*x=1\}\). It is shown that for A doubly stochastic and \(n=3\), W(A) is the convex hull of the point 1 and a certain ellipse or is a line segment or a triangle. For a general value of n some results are given for primitive and cyclic \(n\times n\) doubly stochastic matrices.