Numerical Shadow Rigidity for 3x3 Matrices

The numerical shadow of a complex matrix is the measure obtained by pushing forward that Haar measure on the unit sphere in ${\mathbb{C}}^n$ by the numerical range map. By the choice of measure, unitary equivalence implies same measure, but the reverse is not true for $n\ge 3$. We name a matrix $\mathbf{\text{shadow}}\mathbf{\text{rigid}}$ it its unitary equivalence class and shadow equivalence class coincide. It is known that normal matrices are rigid, but the topic has not been explored beyond this in the literature. The main result of the paper is showing that in the $n=3$ case, a matrix is rigid if and only if it is non-generic. We also show that rigidity is preserved under an affine group action and present three normal-like properties of $3\times 3$ non-generic matrices.