For any square matrix $A$ of size $N$, one defines its numerical range (also called field of values [1]) as a subset of the complex plane: $$ W(A)=\{ z \in \mathbb{C}: z=\bra{\psi} A\ket{\psi}, \ \ket{\psi} \in {\mathbb{C}}^N, \ \braket{\psi}{\psi}=1\}. $$ Following a common convention we denote here the numerical range by $W(X)$, which goes back to the German term “Wertevorrat” originally used by Hausdorff [2].

See section Properties of numerical range.