# Numerical Shadow

The web resource on numerical range and numerical shadow

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numerical-shadow:generalizations:restricted-numerical-shadow:real-numerical-shadow

# Real numerical shadow

## Definition

Real numerical shadow of a matrix $A$ of dimension $d$ is defined as a probability ditribution $P_A(z)$ on the complex plane, supported on the real numerical range $W^\mathbb{R}(A)$. $$P_A(z) := \int_{\Omega} {\rm d} \mu(\psi) \delta\Bigl( z-\langle \psi|A|\psi\rangle\Bigr),$$ where $\mu(\psi)$ denotes the unique unitarily invariant (Fubini-Study) measure on the set $$\Omega=\{\ket{\psi} \in \mathbb{R}^d: \braket{\psi}{\psi}=1\}.$$

## Examples

numerical-shadow/generalizations/restricted-numerical-shadow/real-numerical-shadow.txt · Last modified: 2018/10/08 08:55 by plewandowska