Numerical Shadow

The web resource on numerical range and numerical shadow

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numerical-shadow:examples:2x2

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

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

numerical-shadow/examples/2x2.txt · Last modified: 2013/05/20 00:05 by lpawela