If a matrix satisfies $U+U^T\geq 0$, does its unitarily similar counterpart also satisfy the inequality?

Question

If a real $n\times n$ matrix $U$ satisfies $U+U^T\geq 0$ (i.e., positive semi-definite), does its unitarily similar counterpart $V = W U W^T$, with $WW^T = W^T W = I$, also satisfy $V+V^T \geq 0$?

Answer 1

We have $V+V^{T}=W(U+U^{T})W^{T}.$ It follows that

$$((V+V^{T})x|x)=((U+U^{T})W^{T}x|W^{T}x) \ge 0.$$