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 $A$ 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 $$ \Omega=\{\ket{\psi} \in \mathbb{R}^N: \braket{\psi}{\psi}=1\}. $$

Examples

numerical-shadow/generalizations/restricted-numerical-shadow/real-numerical-shadow.txt · Last modified: 2013/08/03 21:00 by lpawela