The web resource on numerical range and numerical shadow

Hermitian matrices

In the case of Hermitian matrices the numerical shadow is a one dimensional distribution.

Diagonal matrices

Example 1

The matrix is $\mathrm{diag}(0,6)$

Standard numerical shadow with respect to complex states. Numerical shadow with respect to real states.

Non-normal matrices

Example 1

The matrix is $$\begin{pmatrix} 0&1\\ 0&0 \end{pmatrix}.$$ The shadow resembles a physical shadow cast by the hollow sphere made of transparent material when illuminated by a light source at infinity.

Standard numerical shadow with respect to complex states. Cross-section of the shadow along the real axis.

Example 2

The matrix is $$\begin{pmatrix} 0&1\\ 0&1 \end{pmatrix}.$$ The shadow resembles physical shadow cast by the hollow sphere made of transparent material when illuminated by a light source at infinity, but with screen not perpendicular to the light rays.

Standard numerical shadow with respect to complex states. Cross-section of the shadow along the real axis.