With $K = J_{n}$ and $\varphi(X,Y) = X^{\top} J_{n} Y$: Show that for all $(i,j) \in \{1,\ldots,2n\}^{2}$, $$\varphi(e_{i}, e_{j}) = \delta_{i+n,j} - \delta_{i,j+n}$$ (one may start with the case where $(i,j) \in \{1,\ldots,n\}^{2}$ then generalize).

With $K = J_{n}$ and $\varphi(X,Y) = X^{\top} J_{n} Y$: Show that for all $X \in \mathcal{M}_{2n,1}(\mathbb{R})$, $J_{n} X \in X^{\perp}$ and compute $\varphi(J_{n} X, X)$.

With $K = J_{n}$ and $\varphi(X,Y) = X^{\top} J_{n} Y$: If $Y \in \mathcal{M}_{2n,1}(\mathbb{R})$, we denote by $Y^{J_{n}}$ the set of vectors $Z$ of $\mathcal{M}_{2n,1}(\mathbb{R})$ such that $\varphi(Y,Z) = 0$. Show that $X^{J_{n}} = (J_{n} X)^{\perp}$.

With $K = J_{n}$ and $\varphi(X,Y) = X^{\top} J_{n} Y$: Let $P$ be a symplectic and orthogonal matrix whose columns are denoted $X_{1}, \ldots, X_{2n}$. Show that, for all $(i,j) \in \{1,\ldots,2n\}^{2}$, $$\left\{\begin{array}{l} \|X_{i}\| = 1 \\ i \neq j \Longrightarrow X_{i} \perp X_{j} \\ \varphi(X_{i}, X_{j}) = \delta_{i+n,j} - \delta_{i,j+n} \end{array}\right.$$

With $K = J_{n}$, $\varphi(X,Y) = X^{\top} J_{n} Y$, and $P$ a symplectic and orthogonal matrix with columns $X_{1}, \ldots, X_{2n}$ satisfying the properties of Q13: Show that, for all $i \in \{1,\ldots,n\}$, $X_{i}^{J_{n}} = X_{i+n}^{\perp}$.

With $K = J_{n}$, $\varphi(X,Y) = X^{\top} J_{n} Y$, and $P$ a symplectic and orthogonal matrix with columns $X_{1}, \ldots, X_{2n}$ satisfying the properties of Q13: Show that, for all $i \in \{1,\ldots,n\}$, $X_{i+n} = -J_{n} X_{i}$.

Let $$A = \frac{1}{8} \left(\begin{array}{llll} 9 & 1 & 3 & 3 \\ 1 & 9 & 3 & 3 \\ 3 & 3 & 9 & 1 \\ 3 & 3 & 1 & 9 \end{array}\right).$$ Show that $A \in \mathcal{S}_{4}(\mathbb{R}) \cap \mathrm{Sp}_{4}(\mathbb{R})$.

We assume that there exists a symplectic form $\omega$ on $\mathbb { R } ^ { n }$ and we denote by $\Omega \in \mathcal { M } _ { n } ( \mathbb { R } )$ the matrix defined by
$$\Omega = \left( \omega \left( e _ { i } , e _ { j } \right) \right) _ { 1 \leqslant i , j \leqslant n }$$
where $(e _ { 1 } , \ldots , e _ { n })$ denotes the canonical basis of $\mathbb { R } ^ { n }$. Show that
$$\forall ( x , y ) \in \mathbb { R } ^ { n } \times \mathbb { R } ^ { n } , \quad \omega ( x , y ) = X ^ { \top } \Omega Y$$
where $X$ and $Y$ denote the columns of the coordinates of $x$ and $y$ in the canonical basis of $\mathbb { R } ^ { n }$.

We assume that there exists a symplectic form $\omega$ on $\mathbb { R } ^ { n }$ with associated matrix $\Omega = \left( \omega \left( e _ { i } , e _ { j } \right) \right) _ { 1 \leqslant i , j \leqslant n }$, and that $\omega(x,y) = X^{\top} \Omega Y$ for all $x,y \in \mathbb{R}^n$. Deduce that $\Omega$ is antisymmetric and invertible.

We assume that there exists a symplectic form $\omega$ on $\mathbb { R } ^ { n }$ with associated matrix $\Omega$ that is antisymmetric and invertible. Conclude that the integer $n$ is even.

We assume $n = 2m$ and denote by $J \in \mathcal { M } _ { 2m } ( \mathbb { R } )$ the matrix defined in blocks by
$$J = \left( \begin{array} { c c } 0 & - I _ { m } \\ I _ { m } & 0 \end{array} \right)$$
and $j$ the endomorphism of $\mathbb { R } ^ { n }$ canonically associated with $J$. Show that the map $b _ { s } : \left\lvert \, \begin{array} { c c l } \mathbb { R } ^ { n } \times \mathbb { R } ^ { n } & \rightarrow & \mathbb { R } \\ ( x , y ) & \mapsto & \langle x , j ( y ) \rangle \end{array} \right.$ is a symplectic form on $\mathbb { R } ^ { n }$.

Let $A , B , C , D$ be in $\mathcal { M } _ { m } ( \mathbb { R } )$ and let $M = \left( \begin{array} { l l } A & B \\ C & D \end{array} \right) \in \mathcal { M } _ { 2 m } ( \mathbb { R } )$ (block decomposition), where $J = \left( \begin{array}{cc} 0 & -I_m \\ I_m & 0 \end{array} \right)$. Show that $M \in \mathrm { Sp } _ { 2 m } ( \mathbb { R } )$ if and only if
$$A ^ { \top } C \text { and } B ^ { \top } D \text { are symmetric } \quad \text { and } \quad A ^ { \top } D - C ^ { \top } B = I _ { m } .$$

Show that $\mathrm { Sp } _ { 2 } ( \mathbb { R } ) = \mathrm { SL } _ { 2 } ( \mathbb { R } )$.

We denote by $\operatorname { OSp } _ { n } ( \mathbb { R } ) = \operatorname { Sp } _ { n } ( \mathbb { R } ) \cap \mathrm { O } _ { n } ( \mathbb { R } )$ and $\mathcal { C } _ { J } = \left\{ M \in \mathcal { M } _ { n } ( \mathbb { R } ) \mid J M = M J \right\}$. Show that $\operatorname { OSp } _ { n } ( \mathbb { R } ) \subset \mathcal { C } _ { J }$.

Let $M \in \mathrm { Sp } _ { n } ( \mathbb { R } )$ and let $S \in \mathcal { S } _ { n } ( \mathbb { R } )$ be a symmetric matrix with strictly positive eigenvalues such that $S ^ { 2 } = M ^ { \top } M$. Show that $S$ is symplectic.
One may consider a basis of eigenvectors of the endomorphism $s$ of $\mathbb { R } ^ { n }$ canonically associated with $S$, and show that $s$ is a symplectic endomorphism of the standard space $(\mathbb { R } ^ { n } , b _ { s })$.

Let $(E , \omega)$ be a symplectic vector space of dimension $n = 2m$. Let $a \in E$ be a non-zero vector and $\lambda \in \mathbb { R }$ be a real number. Show that the map $\tau _ { a } ^ { \lambda }$ defined by
$$\forall x \in E , \quad \tau _ { a } ^ { \lambda } ( x ) = x + \lambda \omega ( a , x ) a$$
is a transvection of $E$ and that it is a symplectic endomorphism of this same space.

We assume that there exists a symplectic form $\omega$ on $\mathbb { R } ^ { n }$ and we denote by $\Omega \in \mathcal { M } _ { n } ( \mathbb { R } )$ the matrix defined by
$$\Omega = \left( \omega \left( e _ { i } , e _ { j } \right) \right) _ { 1 \leqslant i , j \leqslant n }$$
where $(e _ { 1 } , \ldots , e _ { n })$ denotes the canonical basis of $\mathbb { R } ^ { n }$. Show that
$$\forall ( x , y ) \in \mathbb { R } ^ { n } \times \mathbb { R } ^ { n } , \quad \omega ( x , y ) = X ^ { \top } \Omega Y$$
where $X$ and $Y$ denote the columns of the coordinates of $x$ and $y$ in the canonical basis of $\mathbb { R } ^ { n }$.

We assume $n = 2m$ and denote by $J \in \mathcal { M } _ { n } ( \mathbb { R } )$ the matrix defined in blocks by
$$J = \left( \begin{array} { c c } 0 & - I _ { m } \\ I _ { m } & 0 \end{array} \right)$$
and we denote by $j$ the endomorphism of $\mathbb { R } ^ { n }$ canonically associated with $J$. Show that the map $b _ { s } : \left\lvert \, \begin{array} { c c l } \mathbb { R } ^ { n } \times \mathbb { R } ^ { n } & \rightarrow & \mathbb { R } \\ ( x , y ) & \mapsto & \langle x , j ( y ) \rangle \end{array} \right.$ is a symplectic form on $\mathbb { R } ^ { n }$.

Let $A , B , C , D$ be in $\mathcal { M } _ { m } ( \mathbb { R } )$ and let $M = \left( \begin{array} { l l } A & B \\ C & D \end{array} \right) \in \mathcal { M } _ { 2 m } ( \mathbb { R } )$ (block decomposition). Show that $M \in \mathrm { Sp } _ { 2 m } ( \mathbb { R } )$ if and only if
$$A ^ { \top } C \text { and } B ^ { \top } D \text { are symmetric } \quad \text { and } \quad A ^ { \top } D - C ^ { \top } B = I _ { m } .$$

We consider a matrix $M \in \mathrm { Sp } _ { n } ( \mathbb { R } )$. Let $S \in \mathcal { S } _ { n } ( \mathbb { R } )$ be a symmetric matrix with strictly positive eigenvalues such that $S ^ { 2 } = M ^ { \top } M$. Show that $S$ is symplectic.
One may consider a basis of eigenvectors of the endomorphism $s$ of $\mathbb { R } ^ { n }$ canonically associated with $S$, and show that $s$ is a symplectic endomorphism of the standard space $(\mathbb { R } ^ { n } , b _ { s })$.

Let $(E, \omega)$ be a symplectic vector space of dimension $n = 2m$. Let $a \in E$ be a non-zero vector and $\lambda \in \mathbb { R }$ be a real number. Show that the map $\tau _ { a } ^ { \lambda }$ defined by
$$\forall x \in E , \quad \tau _ { a } ^ { \lambda } ( x ) = x + \lambda \omega ( a , x ) a$$
is a transvection of $E$ and that it is a symplectic endomorphism of this same space.

We denote by $\mathbf{e}$ the matrix of $\mathcal{M}_{n,1}(\mathbb{R})$ whose coefficients are all equal to 1, $P = I_n - \frac{1}{n}\mathbf{e}\cdot\mathbf{e}^T$, and $\Delta_n$ the set of EDM of order $n$.
Let $D$ be a non-zero EDM of order $n$. Show that for all $x \in \operatorname{Vect}(\mathbf{e})^\perp$, we have $$x^T D x \leqslant 0.$$