Let $P$ be a positive semi-definite Hermitian matrix (i.e. $P^\dagger=P$, $x^\dagger Px \geq 0$ for all $x \in \mathbb C^n$). Then the matrix can be decomposed as $P= R+iM$ where $R$ and $M$ are strictly real matrices. Clearly $R$ is symmetric and $M$ is skew-symmetric. I call $R$ the real part of $P$.
I am almost certain that the real part $R$ must be positive semi-definite as well as I cannot find a counter-example even through numerical simulation. Is this assumption correct? Is there a way to prove the positivity of $R$ over $\mathbb C^n$ or does somebody have a counter-example?
The real part of $P$ is $(P + \overline{P})/2$. If $P$ is positive semidefinite, so is $\overline{P}$, because $$x^* \overline{P} x = \overline{ \overline{x}^* P \overline{x}} \ge 0$$ And a convex combination of positive semidefinite matrices is positive semidefine.