I'm looking for a sufficient and necessary constraint for the complex square matrix $A$ such that $\text{tr}(A^n)\geq0$ for $n=1,2,3...$
Up to now, it seems that the eigenvalues of $A$ must come in real numbers or complex conjugate pairs if we want $\text{tr}(A^n)\in\mathbb{R}$ for $n=1,2,3...$ (Whereas, I can't give a proof).
So, if we want $\text{tr}(A^n)\geq 0$ for $n=1,2,3...$, what further constraints do we need to impose on the eigenvalues of $A$?