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 consider a polynomial $P \in \mathbb { R } [ X ]$ annihilating $u$ and a decomposition $P = P _ { 1 } \cdots P _ { r }$, where $r \in \mathbb { N } ^ { * }$ and $P _ { 1 } , \ldots , P _ { r }$ are polynomials pairwise coprime in $\mathbb { R } [ X ]$. We denote $F _ { j } = \operatorname { ker } \left[ P _ { j } ( u ) \right]$ for $j = 1 , \ldots , r$. The notation $F^{\omega}$ is defined in question 7. Show that, for all $j$ and $k$ belonging to $\{ 1 , \ldots , r \}$ and distinct, we have $F _ { k } \subset F _ { j } ^ { \omega }$ and $F _ { k } \subset F _ { j } ^ { \omega _ { 1 } }$.
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 consider a polynomial $P \in \mathbb { R } [ X ]$ annihilating $u$ and a decomposition $P = P _ { 1 } \cdots P _ { r }$, where $r \in \mathbb { N } ^ { * }$ and $P _ { 1 } , \ldots , P _ { r }$ are polynomials pairwise coprime in $\mathbb { R } [ X ]$. We denote $F _ { j } = \operatorname { ker } \left[ P _ { j } ( u ) \right]$ for $j = 1 , \ldots , r$. The notation $F^{\omega}$ is defined in question 7.
Show that, for all $j$ and $k$ belonging to $\{ 1 , \ldots , r \}$ and distinct, we have $F _ { k } \subset F _ { j } ^ { \omega }$ and $F _ { k } \subset F _ { j } ^ { \omega _ { 1 } }$.