Let$f\in End(X)$ for an abelian variety X and let $f'=\phi_H^{-1}\hat{f}\phi_H$ be the usual Rosati involution. An endomorphism of abelian varieties is called symmetric if $f=f'$.

Now I want to show that symmetric endomorphisms have real eigenvalues. This is supposed to be similar to the proof that symmetric matrices only posses real eigenvalues.

Any advice appreciated.