Consider a real number $x > 0$, a real symmetric matrix ${\bf A} = {\bf A}^\mathrm{T}$, and the imaginary unit $i$. I want to show that the real part of $$(x{\bf I} + i{\bf A})^{-1}$$ is positive definite.
It's probably a one-liner with the right approach but I just can't find it. The matrix inversion lemma (Woodbury) didn't help me. I'm confident that the statement is true for a physical reason and after checking it numerically for a million random ${\bf A}$ and tiny $x$-values.
A more general form of the statement (also backed up numerically) is: the real part of $$({\bf D} + i{\bf A})^{-1}$$ is positive definite where ${\bf D} = \mathrm{diag}_n(x_n)$ with all $x_n > 0$.