Describing the Numerical Range and $C$-Numerical Range of Matrices via Their Unitary Orbits and the Joukowsky Transform

We generalise the Elliptical Range Theorem to characterise the numerical range of matrices belonging to a subspace of the space of $3\times 3$ matrices. Using Specht’s Theorem, which characterizes when two matrices are unitarily equivalent, we then provide a novel proof of the Elliptical Range Theorem. Finally, we give an explicit description of the $C$-numerical range for $2\times 2$ matrices and for rank-one matrices.