Let $e = (e_1, \ldots, e_n)$ be a basis of $E$. We denote by $e^* = (e_1^*, \ldots, e_n^*)$ the dual basis of $e$.
a) Show that the matrix of $h$ in the bases $e$ and $e^*$ is: $$\operatorname{mat}(h, e, e^*) = \left(\varphi(e_i, e_j)\right)_{\substack{1 \leq i \leq n \\ 1 \leq j \leq n}}$$ This latter matrix will also be called the matrix of $\varphi$ in the basis $e$ and denoted $\operatorname{mat}(\varphi, e)$.
b) Let $(x,y) \in E^2$. We denote by $X$ and $Y$ the column matrices whose coefficients are the components of $x$ and $y$ in the basis $e$. Show that $\varphi(x,y) = {}^t X \Omega Y$ where $\Omega = \operatorname{mat}(\varphi, e)$ and where ${}^t X$ denotes the row matrix obtained by transposing $X$.

$\mathcal{P}$ denotes the vector space of polynomial functions with complex coefficients. We denote for every pair $(P,Q) \in \mathcal{P}^2$, $$\langle P, Q \rangle = \int_0^{+\infty} e^{-t}\bar{P}(t)Q(t)\,dt.$$
Verify that $\langle \cdot, \cdot \rangle$ defines an inner product on $\mathcal{P}$.

Prove that the map $q \mapsto \widetilde { q }$ is a bijection from the set of quadratic forms on $V$ to the set of symmetric bilinear forms on $V$, where $\widetilde { q } : V \times V \rightarrow \mathbb { K }$ is defined by $( x , y ) \mapsto \widetilde { q } ( x , y ) = \frac { 1 } { 2 } ( q ( x + y ) - q ( x ) - q ( y ) )$.

We define $$\varphi _ { p + 1 } \left( v , v ^ { \prime } \right) = { } ^ { t } V \Delta _ { p + 1 } V ^ { \prime } = v _ { 1 } v _ { 1 } ^ { \prime } - \sum _ { i = 2 } ^ { p + 1 } v _ { i } v _ { i } ^ { \prime }$$ and $$q _ { p + 1 } ( v ) = \varphi _ { p + 1 } ( v , v )$$ Express $\varphi _ { p + 1 } \left( v , v ^ { \prime } \right)$ as a function of $q _ { p + 1 } \left( v + v ^ { \prime } \right)$ and $q _ { p + 1 } \left( v - v ^ { \prime } \right)$.

Show that $\omega ( x , x ) = 0$ for all $\omega \in \mathrm { A } ( E )$ and for all $x \in E$.

Let $\omega \in \mathrm { A } ( E )$ and $\mathcal { B } = \left( b _ { 1 } , \ldots , b _ { n } \right)$ a basis of $E$.
(a) Show that there exists a unique matrix $M \in \mathcal { M } _ { n } ( \mathbb { R } )$, whose coefficients we shall specify, such that for all $( x , y ) \in E ^ { 2 } , \omega ( x , y ) = { } ^ { t } X M Y$ where $X , Y \in \mathbb { R } ^ { n }$ are the column matrices representing respectively $x$ and $y$ in the basis $\mathcal { B }$: $$X = \left( \begin{array} { c } x _ { 1 } \\ \vdots \\ x _ { n } \end{array} \right) , \quad Y = \left( \begin{array} { c } y _ { 1 } \\ \vdots \\ y _ { n } \end{array} \right) , \quad \begin{aligned} & x = x _ { 1 } b _ { 1 } + \cdots + x _ { n } b _ { n } \\ & y = y _ { 1 } b _ { 1 } + \cdots + y _ { n } b _ { n } . \end{aligned}$$ We then denote $M = \operatorname { Mat } _ { \mathcal { B } } ( \omega )$.
(b) Show that $M$ is antisymmetric, that is, ${ } ^ { t } M = - M$.
(c) Show that the vector space $\mathrm { A } ( E )$ is of dimension 1 when $E$ is of dimension 2.
(d) Show the equivalence between the three following statements.
$\left( \mathcal { E } _ { 1 } \right) : \quad \omega$ is a symplectic form on $E$.
$\left( \mathcal { E } _ { 2 } \right) : \quad$ For all $x \in E \backslash \{ 0 \}$, there exists $y \in E$ such that $\omega ( x , y ) \neq 0$.
$\left( \mathcal { E } _ { 3 } \right) : \quad \operatorname { Mat } _ { \mathcal { B } } ( \omega )$ is invertible.

Show that, if there exists a symplectic form on $E$, then $E$ is of even dimension.

Show that the map $\omega _ { 0 }$ defined by $$\begin{array} { r l c c } \omega _ { 0 } : & \mathbb { R } ^ { n } \times \mathbb { R } ^ { n } & \rightarrow & \mathbb { R } \\ ( X , Y ) & \mapsto & { } ^ { t } X J _ { n } Y \end{array}$$ is a symplectic form on $\mathbb { R } ^ { n }$.

We fix a symplectic form $\omega$ on $E$. The purpose of questions 6 to 9 is to show that there exists a basis $\mathcal { B }$ of $E$ such that $\operatorname { Mat_{\mathcal {B}} } ( \omega ) = J _ { n }$.
Treat the case where $E$ is of dimension 2.

We fix a symplectic form $\omega$ on $E$. Let $F$ be a vector subspace of $E$.
(a) Show that, for every linear form $u : F \rightarrow \mathbb { R }$, there exists a linear form $\widetilde { u } : E \rightarrow \mathbb { R }$ whose restriction to $F$ coincides with $u$.
We denote by $F ^ { \omega }$ the vector subspace of $E$ defined by $$F ^ { \omega } = \{ x \in E : \forall y \in F , \omega ( x , y ) = 0 \}$$ and $\psi _ { F }$ the linear map defined by $$\left\lvert \, \begin{aligned} \psi _ { F } : \quad E & \rightarrow F ^ { * } \\ x & \left. \mapsto \varphi _ { \omega } ( x ) \right| _ { F } \end{aligned} \right.$$ where $\left. \varphi _ { \omega } ( x ) \right| _ { F }$ is the restriction of $\varphi _ { \omega } ( x )$ to $F$.
(b) Show that the restriction of $\omega$ to $F \times F$ is a symplectic form on $F$ if and only if $F \cap F ^ { \omega } = \{ 0 \}$.
(c) What are the kernel and image of $\psi _ { F }$ ?
(d) Show that $\operatorname { dim } ( F ) + \operatorname { dim } \left( F ^ { \omega } \right) = \operatorname { dim } ( E )$.
(e) Show that, if the restriction of $\omega$ to $F \times F$ is a symplectic form on $F$, then $E = F \oplus F ^ { \omega }$ and the restriction of $\omega$ to $F ^ { \omega } \times F ^ { \omega }$ is a symplectic form on $F ^ { \omega }$.

We fix a symplectic form $\omega$ on $E$. Conclude that there exists a basis $\mathcal { B }$ of $E$ such that $\operatorname { Mat_{\mathcal {B}} } ( \omega ) = J _ { n }$. Deduce that $\omega$ tames at least one complex structure on $E$.

We fix two symplectic forms $\omega$ and $\omega _ { 1 }$ on $E$. Show that there exists a unique $u \in \mathrm { GL } ( E )$ such that $\omega _ { 1 } ( x , y ) = \omega ( u ( x ) , y )$ for all $( x , y ) \in E ^ { 2 }$. Show then that $u$ belongs to the set $\mathcal { S }$ defined by $$\mathcal { S } = \left\{ u \in \mathrm { GL } ( E ) : \forall ( x , y ) \in E ^ { 2 } , \omega ( x , u ( y ) ) = \omega ( u ( x ) , y ) \right\}$$

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 $z_1, z_2, y_1, y_2 \in E$ be as defined in question 16.
Show that $$\begin{aligned} & \omega \left( z _ { 1 } , z _ { 2 } \right) = \omega \left( y _ { 1 } , y _ { 2 } \right) = 0 \\ & \omega \left( z _ { 1 } , y _ { 1 } \right) = - \omega \left( z _ { 2 } , y _ { 2 } \right) \\ & \omega \left( z _ { 1 } , y _ { 2 } \right) = \omega \left( z _ { 2 } , y _ { 1 } \right) \end{aligned}$$

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 $Z, Y \in \mathbb{C}^4$ be eigenvectors as in question 15, with $z_1, z_2, y_1, y_2$ as defined in question 16.
Show that, by replacing $Y$ with $\xi Y$ where $\xi \in \mathbb { C } \backslash \{ 0 \}$ is suitably chosen, we have $\omega \left( z _ { 1 } , y _ { 1 } \right) = - 1$ and $\omega \left( z _ { 1 } , y _ { 2 } \right) = 0$.

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 $j = 1 , \ldots , r$, and $F_1, \ldots, F_r$ are pairwise orthogonal for $\omega$ and for $\omega_1$ (as shown in question 21).
Deduce that, for all $j \in \{ 1 , \ldots , r \}$, the restrictions of $\omega$ and $\omega _ { 1 }$ to $F _ { j } \times F _ { j }$ are symplectic forms on $F _ { j }$.

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 $E$. We consider the propositions:
$\left( \mathcal { F } _ { 1 } \right) :$ There exists a complex structure tamed by $\omega$ and by $\omega _ { 1 }$.
$\left( \mathcal { F } _ { 2 } \right) :$ The segment $\left[ \omega , \omega _ { 1 } \right] = \left\{ ( 1 - \theta ) \omega + \theta \omega _ { 1 } ; \theta \in [ 0,1 ] \right\}$ is included in the set of symplectic forms on $E$.
Let $u$ be the automorphism of $E$ defined in question 13. Suppose that $\left( \mathcal { F } _ { 2 } \right)$ is satisfied and that the characteristic polynomial of $u$ has roots of multiplicity at most 2 in $\mathbb { C }$. Show that $( \mathcal { F } _ { 1 } )$ is satisfied.
Hint: You may prove and then use the fact that, for all $\theta \in \mathbb { R } \backslash \pi \mathbb { Z }$, there exists $\phi \in \mathbb { R }$ such that, for all $X \in \mathbb { R } ^ { 2 } \backslash \{ 0 \} , { } ^ { t } X R _ { \phi } X > 0$ and ${ } ^ { t } X R _ { \theta + \phi } X > 0$.

In this question only, $n$ is any non-zero natural integer. Determine $J_{n}^{2}$ and show that $J_{n} \in \mathrm{Sp}_{2n}(\mathbb{R}) \cap \mathcal{A}_{2n}(\mathbb{R})$.
Recall: $J_{n} = \left(\begin{array}{cc} 0_{n,n} & I_{n} \\ -I_{n} & 0_{n,n} \end{array}\right)$, and a matrix $M \in \mathcal{M}_{2n}(\mathbb{R})$ is symplectic if and only if $M^{\top} J_{n} M = J_{n}$.

In the case $n=1$: Let $M$ be an orthogonal matrix of size $2 \times 2$. We denote by $M_{1} = \binom{x_{1}}{x_{2}}$ and $M_{2} = \binom{y_{1}}{y_{2}}$ the two columns of $M$. Show the equivalence $$M \text{ is symplectic } \Longleftrightarrow M_{2} = -J_{1} M_{1}.$$

In the case $n=1$: Let $X_{1} \in \mathcal{M}_{2,1}(\mathbb{R})$ have norm 1. Show that the square matrix consisting of columns $X_{1}$ and $-J_{1} X_{1}$ is both orthogonal and symplectic.

In the case $n=1$: Determine the matrices of size $2 \times 2$ that are both antisymmetric and symplectic and show that they are not diagonalizable in $\mathbb{R}$.

Let $K$ be an antisymmetric matrix and $\varphi$ the application from $\left(\mathcal{M}_{2n,1}(\mathbb{R})\right)^{2}$ to $\mathbb{R}$ such that $$\forall (X,Y) \in \left(\mathcal{M}_{2n,1}(\mathbb{R})\right)^{2}, \quad \varphi(X,Y) = X^{\top} K Y.$$ Show that $\varphi$ is a bilinear form on $\mathcal{M}_{2n,1}(\mathbb{R})$.

Let $K$ be an antisymmetric matrix and $\varphi$ the application from $\left(\mathcal{M}_{2n,1}(\mathbb{R})\right)^{2}$ to $\mathbb{R}$ such that $$\forall (X,Y) \in \left(\mathcal{M}_{2n,1}(\mathbb{R})\right)^{2}, \quad \varphi(X,Y) = X^{\top} K Y.$$ By computing $\varphi(X,X)^{\top}$ in two ways, show that $\varphi$ is alternating. Show similarly that $\varphi$ is antisymmetric.

Throughout the rest of the problem, $K = J_{n}$. For all $X = \left(\begin{array}{c} x_{1} \\ x_{2} \\ \vdots \\ x_{2n} \end{array}\right) \in \mathcal{M}_{2n,1}(\mathbb{R})$ and for all $Y = \left(\begin{array}{c} y_{1} \\ y_{2} \\ \vdots \\ y_{2n} \end{array}\right) \in \mathcal{M}_{2n,1}(\mathbb{R})$, where $\varphi(X,Y) = X^{\top} J_{n} Y$, show the equality $$\varphi(X,Y) = \sum_{k=1}^{n} \left(x_{k} y_{k+n} - x_{k+n} y_{k}\right).$$