I want to start saying that this is not a HW problem. I have been reading a paper in Diagonal Riccati Stability and while reading one of the proofs I had an issue understating its conclusion.
Let $S$ be the set of all real, symmetric and negative definite matrices.
Let $H\in \mathrm{sym}(2n,\mathbb{R})$ such that $\forall X\in S$ we have $\mathrm{Tr}(HX)\leq 0$. Then we conclude that $H \geq 0$ (i.e. $H$ is positive semidefinite)
I tried to show that $H$ has to be positive semidefinite using the fact that $H$ is real, symmetric and hence its orthogonally diagonalizable. So we can write $H=UDU'$ where $D$ is diagonal and $U$is orthogonal with $U'U=I$.
But still didn't get it right. Any help appreciated.