Properties of numerical range $W(A)$ of a matrix $A$ of dimension $N$ [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). In particular, we can consider elliptical shape (see [7]),
   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 $A$)
8. if $N=4$:
   1. for $$ A = \left[ \begin{matrix} \alpha \1 & C\\ D & \beta \1 \\ \end{matrix} \right] $$ the numerical range is the convex hull of two non-concentric ellipses [8],

More interesting properties you can find in [9], [10], [11], [12], [13], [14], [15], [16], [17]. In the case of tridiagonal matrices, the characteristic of the numerical range was presented in [18], [19]. For properties of the numerical range in the infinite dimension see [20], [21].

An example application of numerical range can be found in [22]:

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

We define the Gau–Wu number of a matrix $A$, denoted $k(A)$, to be the maximum size of an orthonormal set $\{x_1,\ldots,x_k\} \subset \mathcal{C}^n$ such that the values $\bra{x_j}A\ket{x_j}$ lie on $\partial W(A)$. [23]

The properties of Gau-Wu number was widely studied in [24], [25].

Let $W(A) $ will be the numerical range of matrix $A \in M_n(\mathbb{C})$ and let $$ \mathcal{F}_n = \{ W(A): A \in M_n(\mathbb{C}) \}$$ be the set of all numerical ranges arising in $n$ dimensions.

Let $ A $ be such operator that $rank(A- \lambda \1)=2$ for some $\lambda \in \mathbb{C}$. Then $W(A)$ either is an element of $\mathcal{F}_3$, or the convex hull of two ellipses having a common focus, or has at most one flat portion on its boundary.

More properties (with examples) about rank-two operators is studied in [26].