The web resource on numerical range and numerical shadow

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numerical-range:examples:2x2

### Example 1

• For a Jordan matrix $J=\begin{pmatrix}0&1\\0&0\end{pmatrix}$ the numerical range forms a circular disk of radius one half, $W(J) = D(0,\frac{1}{2})$.

Example of a circular numerical range.

### 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 $M=\begin{pmatrix}1&1\ii\\-1\ii&2\end{pmatrix}$

Numerical range of a normal matrix.

### 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 $M=\begin{pmatrix}1&1\\0&-1\end{pmatrix}$.

Numerical range of a non-normal matrix.