# Numerical Shadow

The web resource on numerical range and numerical shadow

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numerical-shadow:generalizations:restricted-numerical-shadow:product-numerical-shadow

# Product numerical shadow

## Definition

Product numerical shadow of a matrix $A$ of dimension $d$ is defined as a probability distribution $P_A(z)$ on the complex plane, supported on the product numerical range $W^\otimes(A)$. $$P_A(z) := \int_{\Omega} {\rm d} \mu(\psi) \delta\Bigl( z- \bra{\psi} A \ket{\psi} \Bigr),$$ where $\mu(\psi)$ denotes the unique local unitarily invariant (Fubini-Study) measure on the set $$\Omega = \{ \ket{\psi} \in \mathbb{C} ^ d: \ket{\psi} = \bigotimes_{i=1}^d \ket{\phi_i}, \text{ for } i=1,\ldots,d\ \braket{\phi_i}{\phi_i}=1 \text{ and } \ket{\phi_i}\in\mathbb{C}^2 \}$$

## Examples

numerical-shadow/generalizations/restricted-numerical-shadow/product-numerical-shadow.txt · Last modified: 2018/10/08 08:56 by plewandowska