You are looking at the numerical range of a matrix. For any (complex) linear transformation $T$, one defines the numerical range of a matrix as
This is a mapping from the unit sphere to the complex plane. In general, $W(T)$ is a subset of the complex plane. If it is a subset of the real line, then $T$ should be a hermitian operator. Some well known results on $W(T)$ are
- $W(T)$ lies in a disc with radius $||T||$.
- $W(T)$ contains all eigenvalues of $T$.
- $W(T)$ is the convex hull of eigenvalues of $T$.
- $W(T)$ is a closed compact convex set for finite-dimensional $T$.
The last result is the well-known Toeplitz-hausdorff theorem. You can learn more on this beautiful theory by searching for this theorem.