Numerical Shadow

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

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numerical-shadow:generalizations:restricted-numerical-shadow:real-numerical-shadow [2013/08/03 21:00]
lpawela
numerical-shadow:generalizations:restricted-numerical-shadow:real-numerical-shadow [2018/10/08 08:55] (current)
plewandowska [Definition]
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 ===== Definition ===== ===== Definition =====
  
-Real numerical shadow of a matrix $A$ is defined as a probability ditribution $P_A(z)$ on the complex plane, supported on the [[numerical-range:​generalizations:​restricted-numerical-range:​real-numerical-range|real numerical range]] $W^\mathbb{R}(A)$.+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 [[numerical-range:​generalizations:​restricted-numerical-range:​real-numerical-range|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),​ P_A(z) := \int_{\Omega} {\rm d} \mu(\psi) \delta\Bigl( z-\langle \psi|A|\psi\rangle\Bigr),​
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 where $\mu(\psi)$ denotes the unique unitarily invariant (Fubini-Study) measure on the set 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\}.+\Omega=\{\ket{\psi} \in \mathbb{R}^d: \braket{\psi}{\psi}=1\}.
 $$ $$
  
numerical-shadow/generalizations/restricted-numerical-shadow/real-numerical-shadow.txt · Last modified: 2018/10/08 08:55 by plewandowska