A doubly stochastic matrix $A \in \mathbb{R}^{n \times n } $ is a matrix for which the entries are non-negative while the row and column sums are all equal to one $$ \sum_{j=1}^{n} a_{ij} = 1 \text{ for } i=1,\ldots,n $$ and $$ \sum_{i=1}^{n} a_{ij} = 1 \text{ for } j=1,\ldots,n$$

Let A be $ 4\times 4$ doubly stochastic matrix. Then, $\partial W(A)$ consists of elliptical arcs and line segments if and only if $$ \mu := \tr (A) - 1 + \frac{ \tr A^3 - \tr (A^\top A^2 )}{ \tr (A^\top A ) - \tr A^2} $$ is an eigenvalue of A (multiple, if $\mu =1$). If, in addition $$ \tr A - 1 - 3\mu >0 , (\tr A -1 - 3\mu )^2 - \tr(A^\top A) + 1 +2 \frac{\det (A) }{\mu} + \mu^2 > 0 $$

then $W(A)$ also has corner points at $\mu$ and 1, and thus four flat portions on the boundary. Otherwise, 1 is the only corner point of $W(A)$ and $\partial W(A)$ consists of two flat portions and one elliptical arc.

Consider the doubly stochastic matrix : $$ A = \begin{pmatrix} 0&1/3&1/4&5/12\\ 1/3&0&1/2&1/6\\ 1/4&9/32&1/4&1/6\\ 5/12&37/96&0&19/96 \end{pmatrix} $$

Using above theorem, we compute $\mu = -1/3.$ By calculating the characteristic polynomial and computin the conditions from Theorem we have that one of condition is negative, Theorem 1 states that $\partial W(A)$ has two flat portions and one elliptical arc. Indeed, A reduces unitarily to $(1) \oplus A_1$ for some $3\times 3 $ matrix $A_1$, and in Fig. 1 we give $W(A)$, and the horizontal ellipse it contains, $W(A_1)$.

Fig. 1 Numerical range of $A$ with two flat portions and an elliptical arc on the boundary, and the ellipse it contains. The eigenvalues of A are indicated by the points.

Consider the doubly stochastic matrix : $$ A = \begin{pmatrix} 0&1/3&1/4&5/12\\ 1/3&0&1/2&1/6\\ 1/4&1/8&1/4&3/8\\ 5/12&13/24&0&1/24 \end{pmatrix} $$ By coincidence, we again compute $\mu = -1/3$, and though the characteristic polynomial again has $\mu $ as a root. The formulas in inequalities on Theorem evaluate to $7/24$ and $59/576$ respectively, so the number of flat portions is still the same. However, Fig. 2 shows that $W(A)$ is the convex hull of a vertical ellipse and the point 1, as opposed to the horizontal ellipse in the previous example. Indeed, the eigenvalues marked in the graph include a complex conjugate pair.

Fig. 2 Numerical range $A$ with two flat portions and an elliptical arc on the boundary, and the contained vertical ellipse. The eigenvalues of $A$ are indicated by the points.

This section is created based on [1] in which we can find more examples also.