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