Real numerical shadow

Definition

Real numerical shadow of a matrix $A$ of dimension $d$ is defined as a probability ditribution $P_A(z)$ on the complex plane, supported on the real numerical range $W^{\mathbb{R}}(A)$. $P_A(z):={\int }_{\mathrm{\Omega }}\mathrm{d}\mu (\psi )\delta (z-⟨\psi |A|\psi ⟩),$ where $\mu (\psi )$ denotes the unique unitarily invariant (Fubini-Study) measure on the set $\mathrm{\Omega }=\{|\psi ⟩\in {\mathbb{R}}^d:⟨\psi |\psi ⟩=1\}.$

Examples

• Real shadow of 2x2 Hermitian matrices
• Real shadow of 3x3 Hermitian matrices
• Real shadow of 4x4 Hermitian matrices
• Real shadow of 4x4 non-normal matrices