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numerical-range:examples:4x4

### Example 1

A generic matrix $$M=\begin{pmatrix} 1 & 1 & 1 & 1 \\ 0 & \ii & 1 & 1 \\ 0 & 0 & -1 & 1 \\ 0 & 0 & 0 & \ii \end{pmatrix}$$ has an oval–like numerical range $W (M)$.

### Example 2

The matrix $$M=\begin{pmatrix} 1 & 1 & 1 & 1 \\ 0 & \ii & 1 & 1 \\ 0 & 0 & -1 & 1 \\ 0 & 0 & 0 & \ii \end{pmatrix}$$ has a numerical range $W (M)$ with one flat part of the boundary $\partial W$.

### Example 3

The matrix $$M=\begin{pmatrix} 1 & 0 & 0 & 1 \\ 0 & \ii & 0 & 1 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix}$$ has a numerical range $W (M)$ with two flat parts of of the boundary $\partial W$.

### Example 4

The matrix $$M=\begin{pmatrix} 1 & 0 & 0 & 1 \\ 0 & \ii & 1 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -\ii \end{pmatrix}$$ has a numerical range $W (M)$ with two parallel flat parts of of the boundary $\partial W$.

### Example 5

The matrix $$M=\begin{pmatrix} 1 & 0 & 0 & 1 \\ 0 & \ii & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -\ii \end{pmatrix}$$ 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 $$M=\begin{pmatrix} \ii & 0 & -1 & 0 \\ 0 & 0 & -1 & 0 \\ 1 & 1 & 1-\ii & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix}$$ 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 $$M=\begin{pmatrix} 1 & 0 & 1 & 0 \\ 0 & \ii & 0 & 1 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -\ii \end{pmatrix}$$ has a numerical range $W (M)$ with four flat parts of $\partial W$.

### Example 8

The matrix $$M=\begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & \ii & 0 & 1 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -\ii \end{pmatrix}$$ 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 $$M=\begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & \ii & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -\ii \end{pmatrix}$$ has a numerical range $W (M)$ equal to the convex hull of eigenvalues.