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 $(x,y) \in E^2$. We assume that $E$ is of dimension 4 and that $u$ has no real eigenvalue. Let $\mathcal{B}$ be a basis of $E$ such that $\operatorname{Mat}_{\mathcal{B}}(\omega) = J_4$, and let $U \in \mathcal{M}_4(\mathbb{R})$ be the matrix of $u$ in $\mathcal{B}$. Show that $U$ is diagonalizable over $\mathbb { C }$. Deduce that there exist $\lambda \in \mathbb { C } \backslash \mathbb { R }$ and vectors $Z$ and $Y$ of $\mathbb { C } ^ { 4 }$ linearly independent over $\mathbb { C }$ such that $U Z = \lambda Z$ and $U Y = \overline{\lambda} Y$.
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 $(x,y) \in E^2$. We assume that $E$ is of dimension 4 and that $u$ has no real eigenvalue. Let $\mathcal{B}$ be a basis of $E$ such that $\operatorname{Mat}_{\mathcal{B}}(\omega) = J_4$, and let $U \in \mathcal{M}_4(\mathbb{R})$ be the matrix of $u$ in $\mathcal{B}$.
Show that $U$ is diagonalizable over $\mathbb { C }$. Deduce that there exist $\lambda \in \mathbb { C } \backslash \mathbb { R }$ and vectors $Z$ and $Y$ of $\mathbb { C } ^ { 4 }$ linearly independent over $\mathbb { C }$ such that $U Z = \lambda Z$ and $U Y = \overline{\lambda} Y$.