I have a simple question and I want to know if you can give me counterexamples, or maybe it is true. Given matrices $A$ and $B$ such that they do not commute and also it is given that matrix $B$ is PSD and the sum $S=A+B$ is also PSD. Is it then true that $A$ has to be PSD?
When we drop the not commuting constraint, it is easy to find counterexamples, namely $A=-I$ and $B=2I$, $B$ is PSD and $A+B$ also, but $A$ is not PSD.
The question is if we can still find examples when we have the not commuting constraint. Thanks.
Given two positive semi-definite matrices $X$ and $Y$, it is not possible that both $X-Y$ and $Y-X$ are positive semi-definite unless $X=Y$.
If, e.g., $X-Y$ is not positive semi-definite, take $S:=X$ and $B:=Y$ with $A=S-B=X-Y$ to generate a counterexample.