Example 1

A generic matrix $M=\left(\begin{array}{ccccccccccccc}1& 1& 1& 10& \mathrm{i}& 1& 10& 0& -1& 10& 0& 0& \mathrm{i}\end{array}\right)$ has an oval–like numerical range $W(M)$.

Example 2

The matrix $M=\left(\begin{array}{ccccccccccccc}1& 1& 1& 10& \mathrm{i}& 1& 10& 0& -1& 10& 0& 0& \mathrm{i}\end{array}\right)$ has a numerical range $W(M)$ with one flat part of the boundary $\partial W$.

Example 3

The matrix $M=\left(\begin{array}{ccccccccccccc}1& 0& 0& 10& \mathrm{i}& 0& 10& 0& -1& 00& 0& 0& 1\end{array}\right)$ has a numerical range $W(M)$ with two flat parts of of the boundary $\partial W$.

Example 4

The matrix $M=\left(\begin{array}{ccccccccccccc}1& 0& 0& 10& \mathrm{i}& 1& 00& 0& -1& 00& 0& 0& -\mathrm{i}\end{array}\right)$ has a numerical range $W(M)$ with two parallel flat parts of of the boundary $\partial W$.

Example 5

The matrix $M=\left(\begin{array}{ccccccccccccc}1& 0& 0& 10& \mathrm{i}& 0& 00& 0& -1& 00& 0& 0& -\mathrm{i}\end{array}\right)$ has a numerical range $W(M)$ with three flat parts of $\partial W$ connected with corners and one oval–like part.

Example 6

The matrix $M=\left(\begin{array}{ccccccccccccc}\mathrm{i}& 0& -1& 00& 0& -1& 01& 1& 1-\mathrm{i}& 00& 0& 0& 1\end{array}\right)$ has a numerical range $W(M)$ with three flat parts of $\partial W$ with only one corner and two oval–like parts.

Example 7

The matrix $M=\left(\begin{array}{ccccccccccccc}1& 0& 1& 00& \mathrm{i}& 0& 10& 0& -1& 00& 0& 0& -\mathrm{i}\end{array}\right)$ has a numerical range $W(M)$ with four flat parts of $\partial W$.

Example 8

The matrix $M=\left(\begin{array}{ccccccccccccc}1& 0& 0& 00& \mathrm{i}& 0& 10& 0& -1& 00& 0& 0& -\mathrm{i}\end{array}\right)$ has a numerical range $W(M)$ pair of flat parts of $\partial W$ connected with a corner connected with two oval–like parts.

Example 9

The matrix $M=\left(\begin{array}{ccccccccccccc}1& 0& 0& 00& \mathrm{i}& 0& 00& 0& -1& 00& 0& 0& -\mathrm{i}\end{array}\right)$ has a numerical range $W(M)$ equal to the convex hull of eigenvalues.