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. Let $\mathcal { B }$ be a basis of $E$ such that $\operatorname { Mat } _ { \mathcal { B } } ( \omega ) = J _ { 4 }$. Let $U \in \mathcal { M } _ { 4 } ( \mathbb { R } )$ be the matrix of $u$ in the basis $\mathcal { B }$. (a) What relation is there between the matrices $J _ { 4 }$ and $U$ ? (b) Show that there exist $N \in \mathcal { M } _ { 2 } ( \mathbb { R } )$ and $\alpha , \beta \in \mathbb { R }$ such that $$U = \left( \begin{array} { c c } N & \alpha J _ { 2 } \\ \beta J _ { 2 } & { } ^ { t } N \end{array} \right)$$ (c) Determine, as a function of $N , \alpha$ and $\beta$, the coefficients of the polynomial $T$ defined by $T ( X ) = \operatorname { det } \left( N - X I _ { 2 } \right) + \alpha \beta$. Show that $T$ is an annihilating polynomial of $U$.
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. Let $\mathcal { B }$ be a basis of $E$ such that $\operatorname { Mat } _ { \mathcal { B } } ( \omega ) = J _ { 4 }$. Let $U \in \mathcal { M } _ { 4 } ( \mathbb { R } )$ be the matrix of $u$ in the basis $\mathcal { B }$.
(a) What relation is there between the matrices $J _ { 4 }$ and $U$ ?
(b) Show that there exist $N \in \mathcal { M } _ { 2 } ( \mathbb { R } )$ and $\alpha , \beta \in \mathbb { R }$ such that
$$U = \left( \begin{array} { c c } N & \alpha J _ { 2 } \\ \beta J _ { 2 } & { } ^ { t } N \end{array} \right)$$
(c) Determine, as a function of $N , \alpha$ and $\beta$, the coefficients of the polynomial $T$ defined by $T ( X ) = \operatorname { det } \left( N - X I _ { 2 } \right) + \alpha \beta$. Show that $T$ is an annihilating polynomial of $U$.