For any square matrix $A$ of dimension $d$, one defines its numerical shadow as a probability distribution $P_A(z)$ on the complex plane, supported in the numerical range $W(A)$, $$ P_A(z) := \int_{\Omega_d} {\rm d} \mu(\psi) \delta\Bigl( z-\langle \psi|A|\psi\rangle\Bigr) . $$ Here $\mu(\psi)$ denotes the unique unitarily invariant (Fubini-Study) measure on the set $\Omega_d$ of $N$-dimensional pure quantum states. In other words the shadow $P$ of matrix $A$ at a given point $z$ characterizes the likelihood that the expectation value of $A$ among a random pure state is equal to $z$.

Other names for numerical shadow found in literature:

• numerical measure of a complex matrix [1]
• microcanonical distribution [2], [3], [4], [5]