If a real $n\times n$ matrix $U$ satisfies $U+U^T \geq 0$, i.e., positive semidefinite, does its similar counterpart $V = W U W^{-1}$ also satisfy $V+V^T \geq 0$?
This is true for unitary similarity (see my earlier question: If a matrix satisfies $U+U^T\geq 0$, does its unitarily similar counterpart also satisfy the inequality?). I am not sure about the case of $V=WUW^{-1}$.
If you work with simple structures like triangular matrices, you can come up with easy counterexamples.
$D=\left[\begin{matrix}2 &0\\ 0 &1\end{matrix}\right]$
is positive (semi)definite but
$V=\left[\begin{matrix}2 &-100\\ 0 &1\end{matrix}\right]$
is not positive (semi)definite yet is similar to $D$
When you work over reals, orthogonal similarity preserves positive (semi)definiteness because it is an instance of a congruence transform. In general similarity transforms are not congruence transforms and have no reason to preserve something like positive (semi)definiteness.
