Let $A$ and $B$ be two arbitrary matrices of $\mathcal{M}_d(\mathbf{R})$. Using the results of questions 8 and 9, deduce that $$\exp(A+B) = \lim_{n \rightarrow +\infty} \left(\exp\left(\frac{A}{n}\right)\exp\left(\frac{B}{n}\right)\right)^n$$

Let $T$ be a strictly positive real number. We denote by $E(T)$ the set consisting of pairs $(u,v)$ of continuous functions on $[0,T]$ with real values.
A Carnot path controlled by $(u,v) \in E(T)$ is a map $\gamma : [0,T] \rightarrow \mathcal{M}_3(\mathbf{R})$ of class $C^1$ solution of the matrix differential equation: $$\left\{\begin{array}{l} \gamma'(t) = u(t)\gamma(t)M_{1,0,0} + v(t)\gamma(t)M_{0,1,0} \\ \gamma(0) = I_3 \end{array}\right.$$ where $M_{1,0,0}$ and $M_{0,1,0}$ are as defined in the first part.
(a) For all $(u,v) \in E(T)$, justify the existence of a unique Carnot path controlled by $(u,v)$.
(b) Show that $\gamma$ satisfies $$\forall t \in [0,T], \quad \gamma(t) \in \mathbf{H}$$ and explicitly calculate, as a function of $t$, $u$ and $v$, the functions $p(t)$, $q(t)$ and $r(t)$ such that $$\gamma(t) = \exp\left(M_{p(t),q(t),r(t)}\right).$$

For all $(\theta, \varphi) \in \mathbf{R}^2$ and $t \in \mathbf{R}$, we define the controls $$u_{\theta,\varphi}(t) = \sin(\theta - \varphi t) \quad \text{and} \quad v_{\theta,\varphi}(t) = \cos(\theta - \varphi t)$$ and we denote $\gamma_{\theta,\varphi}(t) = \exp\left(M_{p(t),q(t),r(t)}\right)$ the Carnot path controlled by $(u_{\theta,\varphi}, v_{\theta,\varphi})$.
(a) We assume $\varphi \neq 0$. Calculate $p(t)$ and $q(t)$ and verify that $$r(t) = \frac{t\varphi - \sin(t\varphi)}{2\varphi^2}$$
(b) Similarly calculate $\gamma_{\theta,0}(t)$.

The Carnot sphere is the set: $$B(1) = \left\{(p,q,r) \in \mathbf{R}^3 \mid \exists (\theta,\varphi) \in [-\pi,\pi] \times [-2\pi,2\pi], \quad \gamma_{\theta,\varphi}(1) = \exp\left(M_{p,q,r}\right)\right\}.$$
We define the functions $f$ and $g$ on $]0, 2\pi]$ by: $$f(s) = \frac{2(1-\cos s)}{s^2} \quad \text{and} \quad g(s) = \frac{s - \sin s}{2s^2}$$
Show that $f$ and $g$ extend by continuity to $[0, 2\pi]$; that $f$ is then a continuous bijection from $[0, 2\pi]$ onto a set to be specified; and that $g$ attains its maximum at $\pi$.

(a) Show that $O _ { n } ( \mathbb { R } )$ is a subgroup of the group $\mathrm { GL } _ { n } ( \mathbb { R } )$ of invertible matrices.
(b) Show that $O _ { n } ( \mathbb { R } )$ is a compact subset of $\mathcal { M } _ { n } ( \mathbb { R } )$.

Let $M$ and $N$ be in $S _ { n } ( \mathbb { R } )$. Show that there exists $U \in O _ { n } ( \mathbb { R } )$ such that $N = U M U ^ { - 1 }$, if and only if $\chi _ { M } = \chi _ { N }$.

Let $\hat { \lambda } = \left( \lambda _ { 1 } \geqslant \cdots \geqslant \lambda _ { n + 1 } \right) \in \mathbb { R } ^ { n + 1 }$ and $\widehat { \mu } = \left( \mu _ { 1 } \geqslant \cdots \geqslant \mu _ { n } \right) \in \mathbb { R } ^ { n }$. Let $x \in \mathbb { R }$. Form $$\widehat { \lambda } ^ { \prime } = \left( \lambda _ { 1 } \geqslant \cdots \geqslant \lambda _ { i } \geqslant x > \lambda _ { i + 1 } \geqslant \cdots \geqslant \lambda _ { n + 1 } \right)$$ by choosing the integer $i \in \{ 0 , \ldots , n + 1 \}$ appropriately. If $x > \lambda _ { 1 }$, we thus have $i = 0$, while if $x \leqslant \lambda _ { n + 1 }$, we have $i = n + 1$. Similarly form $$\widehat { \mu } ^ { \prime } = \left( \mu _ { 1 } \geqslant \cdots \geqslant \mu _ { j } \geqslant x > \mu _ { j + 1 } \geqslant \cdots \geqslant \mu _ { n } \right) .$$ Assume that $\widehat { \lambda }$ and $\widehat { \mu }$ are interlaced. Show that $j \leqslant i \leqslant j + 1$. By examining each of the two cases $j = i$ or $i - 1$, show that $\widehat { \lambda } ^ { \prime }$ and $\widehat { \mu } ^ { \prime }$ are interlaced.

Let $\widehat { \mu } = \left( \mu _ { 1 } > \cdots > \mu _ { n } \right) \in \mathbb { R } ^ { n }$. We define the polynomials $$Q _ { 0 } = \prod _ { k = 1 } ^ { n } \left( X - \mu _ { k } \right) \quad \text { and } \quad \forall j \in \{ 1 , \ldots , n \} , \quad P _ { j } = \frac { Q _ { 0 } } { \left( X - \mu _ { j } \right) } .$$ (a) Show that the family $\left( Q _ { 0 } , P _ { 1 } , P _ { 2 } , \ldots , P _ { n } \right)$ is a basis of $\mathbb { R } _ { n } [ X ]$.
(b) Let $j \in \{ 1 , \ldots , n \}$. Verify that $( - 1 ) ^ { j - 1 } P _ { j } \left( \mu _ { j } \right) > 0$.

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 $E$. Show by induction that there exists a basis $\widetilde { \mathcal { B } }$ of $E$ such that $$\operatorname { Mat } _ { \widetilde { \mathcal { B } } } ( \omega ) = \left( \begin{array} { c c c c } J _ { 2 } & 0 & \cdots & 0 \\ 0 & J _ { 2 } & \ddots & \vdots \\ \vdots & \ddots & \ddots & 0 \\ 0 & \cdots & 0 & J _ { 2 } \end{array} \right)$$

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. 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 $U$.

What can be said about a nilpotent endomorphism of index 1?

We assume that $n = 2$. Let $u$ be an endomorphism of $E$ nilpotent of index $p \geqslant 2$.
Show that there exists a vector $x$ in $E$ such that $u^{p-1}(x) \neq 0$.

We assume that $n = 2$. Let $u$ be an endomorphism of $E$ nilpotent of index $p \geqslant 2$.
Verify that the family $\left(u^{k}(x)\right)_{0 \leqslant k \leqslant p-1}$ is free. Deduce that $p = 2$.

We assume that $n = 2$. Let $u$ be an endomorphism of $E$ nilpotent of index $p \geqslant 2$.
Show that $\operatorname{Ker} u = \operatorname{Im} u$.

We assume that $n = 2$. Let $u$ be an endomorphism of $E$ nilpotent of index $p \geqslant 2$.
Construct a basis of $E$ in which the matrix of $u$ is equal to $J_2$.

Let $A$ denote a matrix in $\mathcal{M}_n(\mathbb{C})$.
Show that, if $A$ is nilpotent, then 0 is the unique eigenvalue of $A$.

Verify that $\delta$ is a neutral element for the operation $*$.

Deduce that $*$ is commutative.

What can be said about $(\mathbb{A}, +, *)$?

What can be said about the set $\mathcal{M}$ equipped with the operation $*$?