Example 1

• For a Jordan matrix $J=\left(\begin{array}{cc}0& 1\\ 0& 0\end{array}\right)$ the numerical range forms a circular disk of radius one half, $W(J)=D(0,)$.

Example 2

• Normal matrix of order $d=2$: the disk reduces to an interval $[{\lambda }_1,{\lambda }_2]$ in the complex plane. The matrix in the example is $M=\left(\begin{array}{cc}1& 1\mathrm{i}\\ -1\mathrm{i}& 2\end{array}\right)$

Example 3

• Generic matrix of order $d=2$ the numerical range $W(M)$ is an ellipse. The foci of the ellipse are located in the eigenvalues. The matrix in the example is $M=\left(\begin{array}{cc}1& 1\\ 0& -1\end{array}\right)$.