Properties of numerical range $W(A)$ of a matrix $A$ of dimension $d$ [1], [2]:

1. $W(A)$ is a compact subset of $\mathbb{C}$,
2. $W(A)$ is convex set (Hausdorff-Toeplitz theorem) [3], [4],
3. $W(A)$ contains the spectrum of $A$,
4. if $AA^\dagger=A^\dagger A$ ($A$ is a normal matrix) then $W(A)$ is convex hull of spectrum of $A$,
5. if $A=A^\dagger$ then $W(A)=[\lambda_1, \lambda_N]$ forms an interval in the real axis,
6. if $N=2$ then $W(A)$ forms an elliptic disk with eigenvalues $\lambda_1$ and $\lambda_2$ as focal points and the minor axis $d = \sqrt{\tr(AA^\dagger) - |\lambda_1|^2- |\lambda_2|^2}$ [5].
7. if $N=3$ we distinguish four cases (classification by Keeler, Rodman, Spitkovsky) [6]:
   1. $W(A)$ is a compact set of an 'ovular' shape containing three eigenvalues (the generic case),
   2. $W(A)$ is a compact set with one flat part (e.g. convex hull of a cardioid),
   3. $W(A)$ is a compact set with two flat parts (e.g. convex hull of an ellipse and a point outside it),
   4. $W(A)$ if a triangle with eigenvalues in its corners (in the case of normal $A$)

An example application of numerical range can be found in [7]:

For a list of examples, see examples of numerical range.