We fix two symplectic forms $\omega$ and $\omega _ { 1 }$ on $E$, and let $u \in \mathrm{GL}(E)$ be the unique automorphism such that $\omega_1(x,y) = \omega(u(x),y)$ for all $(x,y) \in E^2$. Suppose that the characteristic polynomial of $u$ has roots of multiplicity at most 2 in $\mathbb { C }$. Show that $E$ is the direct sum of subspaces of dimension 2 or 4, pairwise orthogonal for $\omega$ and $\omega _ { 1 }$, and on which the restrictions of $\omega$ and $\omega _ { 1 }$ are symplectic forms.
We fix two symplectic forms $\omega$ and $\omega _ { 1 }$ on $E$, and let $u \in \mathrm{GL}(E)$ be the unique automorphism such that $\omega_1(x,y) = \omega(u(x),y)$ for all $(x,y) \in E^2$.
Suppose that the characteristic polynomial of $u$ has roots of multiplicity at most 2 in $\mathbb { C }$. Show that $E$ is the direct sum of subspaces of dimension 2 or 4, pairwise orthogonal for $\omega$ and $\omega _ { 1 }$, and on which the restrictions of $\omega$ and $\omega _ { 1 }$ are symplectic forms.