Let $u \in \operatorname { Symp } _ { \omega } ( E )$ be a symplectic endomorphism of $E$. Let $\lambda , \mu$ be real eigenvalues of $u$, and let $E _ { \lambda } ( u ) , E _ { \mu } ( u )$ be the associated eigenspaces. Show that, if $\lambda \mu \neq 1$, then the subspaces $E _ { \lambda } ( u )$ and $E _ { \mu } ( u )$ are $\omega$-orthogonal, that is:
$$\forall x \in E _ { \lambda } ( u ) , \quad \forall y \in E _ { \mu } ( u ) , \quad \omega ( x , y ) = 0$$