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Real numerical range

Definition

Real numerical range $W^\mathbb{R}(A)$ of a matrix $A$ of dimension $d$ is a subset of the complex plane defined as $$W^\mathbb{R}(A)=\{z:z=\bra{\psi}A\ket{\psi},\ket{\psi}\in\mathbb{R}^d,\braket{\psi}{\psi}=1\}.$$

Conditions for the generalized numerical range to be real

Fact

The classical numerical range satisfies $$W(CU^\dagger AU) \subset \mathbb{R}$$ for all unitary $U$ if and only if at least one of $C$ and $A$ is scalar and their product is hermitian [1].

1. Marvin Marcus, Markus Sandy, 1985. Conditions for the generalized numerical range to be real. Linear algebra and its applications, 71, Elsevier, pp.219–239.