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

Let $A$ be a real antisymmetric and nilpotent matrix. Show that $A ^ { \top } A = 0 _ { n }$, then that $A = 0 _ { n }$.

We denote $C = \left\{ U \in \mathcal{M}_{n,1}(\mathbb{R}) \mid U^\top U = 1 \right\}$. Show that $\rho(A) \leqslant \max_{U \in C} \left| U^\top A U \right|$.

In this part, $\mathbf{K} = \mathbf{R}$. If $G$ is a closed subgroup of $\mathrm{GL}_n(\mathbf{R})$, we introduce its Lie algebra: $$\mathcal{A}_G = \left\{ M \in \mathcal{M}_n(\mathbf{R}) \mid \forall t \in \mathbf{R} \quad e^{tM} \in G \right\}.$$ In questions 11) to 14), $G$ is an arbitrary closed subgroup of $\mathrm{GL}_n(\mathbf{R})$.
$\mathbf{12}$ ▷ Let $A \in \mathcal{A}_G$ and $B \in \mathcal{A}_G$. Show that the application $$\begin{aligned} u : \mathbf{R} & \longrightarrow \mathcal{M}_n(\mathbf{R}) \\ t & \longmapsto u(t) = e^{tA} \cdot B \cdot e^{-tA} \end{aligned}$$ takes values in $\mathcal{A}_G$.

Let $e = (e_1, \ldots, e_d)$ be an orthonormal basis of $E$, $p$ an integer such that $1 \leqslant p \leqslant d$ and $\mathcal{I}_p = \{\alpha = (i_1, \ldots, i_p) \in \mathbb{N}^p \mid 1 \leqslant i_1 < \cdots < i_p \leqslant d\}$. For all $\alpha = (i_1, \ldots, i_p) \in \mathcal{I}_p$, we denote $e_{\alpha} = (e_{i_1}, \ldots, e_{i_p}) \in E^p$ and for all $\omega$ and $\omega^{\prime}$ elements of $\mathscr{A}_p(E, \mathbb{R})$ $$\langle\omega, \omega^{\prime}\rangle = \sum_{\alpha \in \mathcal{I}_p} \omega(e_{\alpha}) \omega^{\prime}(e_{\alpha}).$$
Show that the inner product $(\omega, \omega^{\prime}) \mapsto \langle\omega, \omega^{\prime}\rangle$ defined above depends only on the inner product on $E$ and not on the choice of the orthonormal basis $e$.

Let $e = (e_1, \ldots, e_d)$ be an orthonormal basis of $E$, $p$ an integer such that $1 \leqslant p \leqslant d$ and $\mathcal{I}_p = \{\alpha = (i_1, \ldots, i_p) \in \mathbb{N}^p \mid 1 \leqslant i_1 < \cdots < i_p \leqslant d\}$. For all $\omega$ and $\omega^{\prime}$ elements of $\mathcal{A}_p(E, \mathbb{R})$: $$\langle\omega, \omega^{\prime}\rangle = \sum_{\alpha \in \mathcal{I}_p} \omega(e_{\alpha}) \omega^{\prime}(e_{\alpha}) \tag{1}$$
Show that the inner product $(\omega, \omega^{\prime}) \mapsto \langle\omega, \omega^{\prime}\rangle$ defined by (1) depends only on the inner product on $E$ and not on the choice of the orthonormal basis $e$.

Let $k \in \mathbb{N}^*, (p_1, \ldots, p_k) \in (\mathbb{R}^d)^k$ and $(b_1, \ldots, b_k) \in \mathbb{R}^k$ such that $$A := \left\{x \in \mathbb{R}^d : p_i \cdot x \leqslant b_i, i = 1, \ldots, k\right\}$$ is non-empty. Show that $A$ is convex and closed. Let $x \in A$, let $I(x) := \{i \in \{1, \ldots, k\} : p_i \cdot x = b_i\}$, show that $$x \in \operatorname{Ext}(A) \Longleftrightarrow \operatorname{rank}\left(\{p_i, i \in I(x)\}\right) = d$$ deduce that $\operatorname{Ext}(A)$ is a finite set (possibly empty) whose cardinality is at most $2^k$.

Let $k \in \mathbb{N}^*, (p_1, \ldots, p_k) \in (\mathbb{R}^d)^k$ and $(b_1, \ldots, b_k) \in \mathbb{R}^k$ such that $$A := \left\{x \in \mathbb{R}^d : p_i \cdot x \leqslant b_i, i = 1, \ldots, k\right\}$$ is non-empty. Show that $A$ is convex and closed. Let $x \in A$, let $I(x) := \left\{i \in \{1, \ldots, k\} : p_i \cdot x = b_i\right\}$, show that $$x \in \operatorname{Ext}(A) \Longleftrightarrow \operatorname{rank}\left(\{p_i, i \in I(x)\}\right) = d$$ deduce that $\operatorname{Ext}(A)$ is a finite set (possibly empty) whose cardinality is at most $2^k$.

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.

Given that any symplectic form on $\mathbb{R}^n$ has an associated matrix $\Omega$ that is antisymmetric and invertible, conclude that the integer $n$ is even.

Suppose $n \geqslant 3$. Give an example of a matrix in $\mathcal { M } _ { n } ( \mathbb { R } )$ with zero trace and zero determinant, but not nilpotent.

Let $A \in \mathcal{S}_n(\mathbb{R})$ and $C = \left\{ U \in \mathcal{M}_{n,1}(\mathbb{R}) \mid U^\top U = 1 \right\}$. Prove that $\rho(A) = \max_{U \in C} \left| U^\top A U \right|$.

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 $\mathbb { R } ^ { n }$.

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 $\left( E _ { 1 } , \ldots , E _ { n } \right)$ be the canonical basis of $\mathcal { M } _ { n , 1 } ( \mathbb { R } )$. We denote $V = \sum _ { k = 1 } ^ { n } E _ { k }$.
For $i \in \llbracket 1 , n \rrbracket$, express $E _ { i }$ in terms of $V$ and of $V - 2 E _ { i }$. Deduce that $\mathcal { M } _ { n , 1 } ( \mathbb { R } ) = \operatorname { Vect } \left( \mathcal { V } _ { n , 1 } \right)$.

Let $A \in \mathcal{S}_n(\mathbb{R})$ and $C = \left\{ U \in \mathcal{M}_{n,1}(\mathbb{R}) \mid U^\top U = 1 \right\}$. We further assume that the eigenvalues of $A$ are all positive. Show then that $\rho(A) = \max_{U \in C} \left( U^\top A U \right)$.

$\mathbf{16}$ ▷ Show that the differential at the point $I_n$ of the application $\det: \mathcal{M}_n(\mathbf{R}) \rightarrow \mathbf{R}$ is the linear form ``trace''.

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
$$A ^ { \top } C \text { and } B ^ { \top } D \text { are symmetric } \quad \text { and } \quad A ^ { \top } D - C ^ { \top } B = I _ { m } .$$

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 $n$ discrete random variables $Y_1, \ldots, Y_n$ with random vector $Y$ and covariance matrix $\Sigma_Y$. We denote by $P$ the change of basis matrix from the canonical basis of $\mathcal{M}_{n,1}(\mathbb{R})$ to an orthonormal basis formed by eigenvectors of $\Sigma_Y$. We define the discrete random variable $X = P^\top Y = \left(\begin{array}{c} X_1 \\ \vdots \\ X_n \end{array}\right)$.
Prove that $\Sigma_X$ is a diagonal matrix.

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

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

We denote by $\mathcal { C } _ { J } = \left\{ M \in \mathcal { M } _ { n } ( \mathbb { R } ) \mid J M = M J \right\}$ the centralizer of the matrix $J = \left( \begin{array}{cc} 0 & -I_m \\ I_m & 0 \end{array} \right)$. Show that, for every matrix $M \in \mathcal { M } _ { n } ( \mathbb { R } ) \left( = \mathcal { M } _ { 2 m } ( \mathbb { R } ) \right)$,
$$M \in \mathcal { C } _ { J } \quad \Longleftrightarrow \quad \exists ( U , V ) \in \mathcal { M } _ { m } ( \mathbb { R } ) \times \mathcal { M } _ { m } ( \mathbb { R } ) , \quad M = \left( \begin{array} { c c } U & - V \\ V & U \end{array} \right) .$$

We denote by $\mathcal { C } _ { J } = \left\{ M \in \mathcal { M } _ { n } ( \mathbb { R } ) \mid J M = M J \right\}$ the centralizer of the matrix $J$. Show that, for every matrix $M \in \mathcal { M } _ { n } ( \mathbb { R } ) \left( = \mathcal { M } _ { 2 m } ( \mathbb { R } ) \right)$,
$$M \in \mathcal { C } _ { J } \quad \Longleftrightarrow \quad \exists ( U , V ) \in \mathcal { M } _ { m } ( \mathbb { R } ) \times \mathcal { M } _ { m } ( \mathbb { R } ) , \quad M = \left( \begin{array} { c c } U & - V \\ V & U \end{array} \right) .$$

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 $\operatorname { OSp } _ { n } ( \mathbb { R } ) \subset \mathcal { C } _ { J }$.