I have arrived at this result from a very different perspective (quantum operations) but, being a completely algebraic result, I was hoping that there would be a simple algebraic way of looking at it too.
Let $P$ be a positive semidefinite matrix. Let $E$ be a diagonal matrix with real entries such that -
- Tr$(E)=0$
- Diag$(P+E) \succeq 0 $ [That is, for all $i$ , $P_{ii}+E_{ii}\geq 0$]
Prove that $P+E$ is positive semidefinite.
Thanks in advance!