Let $\mathcal{U}_q$ be the set of endomorphisms $\phi \in \mathcal{L}(V)$ that are compatible with $P_a$.
20a. Show that there exists a unique algebra morphism $\Psi_a : \mathcal{U}_q \rightarrow \mathcal{L}(W_{\ell})$ such that $$\forall \phi \in \mathcal{U}_q, \quad \Psi_a(\phi) \circ P_a = P_a \circ \phi$$
20b. Show that $\phi \in \mathcal{U}_q$ is contained in the kernel of $\Psi_a$ if and only if the image of $\phi$ is in the subspace of $V$ spanned by the vectors $v_i - a^p v_r, i \in \mathbf{Z}$, where $i = p\ell + r$ is the Euclidean division of $i$ by $\ell$.

Let $\mathcal{U}_q$ be the set of endomorphisms $\phi \in \mathcal{L}(V)$ that are compatible with $P_a$, and $\Psi_a : \mathcal{U}_q \rightarrow \mathcal{L}(W_{\ell})$ the unique algebra morphism such that $\Psi_a(\phi) \circ P_a = P_a \circ \phi$ for all $\phi \in \mathcal{U}_q$. We study $\Psi_a(E)$ in this question.
21a. Determine $\Psi_a(E)(v_0)$.
21b. Deduce $\Psi_a(E^{\ell})$.
21c. Calculate the dimension of the vector subspace $\mathbf{C}[\Psi_a(E)]$.
21d. Calculate the eigenvectors of $\Psi_a(E)$.

Let $\mathcal{U}_q$ be the set of endomorphisms $\phi \in \mathcal{L}(V)$ that are compatible with $P_a$, and $\Psi_a : \mathcal{U}_q \rightarrow \mathcal{L}(W_{\ell})$ the unique algebra morphism such that $\Psi_a(\phi) \circ P_a = P_a \circ \phi$ for all $\phi \in \mathcal{U}_q$. Let $W$ be a non-zero subspace of $W_{\ell}$ stable under $\Psi_a(H)$.
22a. Show that $W$ contains at least one of the vectors $v_i$.
22b. What can be said if $W$ is moreover stable under $\Psi_a(E)$?

Let $A$ be a real square matrix of size $n$ and $b$ an element of $\mathbb{R}^n$. Let $f$ be the map from $\mathbb{R}^n$ to $\mathbb{R}^n$ defined by $$\forall x \in \mathbb{R}^n \quad f(x) = Ax + b$$ Show that $f$ is of class $C^1$ and specify its Jacobian matrix $J_f(x)$ at every point $x$ of $\mathbb{R}^n$.

We denote by $A$ a real square matrix of size 2 and we set, for all $x$ in $\mathbb{R}^2$, $f(x) = Ax$.
For $x$ in $\mathbb{R}^2$, express $\operatorname{div}_f(x)$ using only $A$.

Let $f$ be a function of class $C^2$ from $\mathbb{R}^n$ to itself. We consider the proposition $(\mathcal{P})$: for all $x$ in $\mathbb{R}^n$, the Jacobian matrix $J_f(x)$ of $f$ is orthogonal.
For $x$ in $\mathbb{R}^n$ and $i$, $j$, $k$ in $\llbracket 1, n \rrbracket$, we denote $$\alpha_{i,j,k}(x) = \sum_{p=1}^n \frac{\partial f_p}{\partial x_i}(x) \cdot \frac{\partial^2 f_p}{\partial x_j \partial x_k}(x)$$
We assume $(\mathcal{P})$. Show that there exist an orthogonal matrix $A$ and an element $b$ of $\mathbb{R}^n$ such that, for all $x$ in $\mathbb{R}^n$, $f(x) = Ax + b$.
One may interpret the relations $\alpha_{i,j,k} = 0$ using matrix products.

Let $f$ be a function of class $C^2$ from $\mathbb{R}^n$ to itself. We consider the proposition $(\mathcal{P})$: for all $x$ in $\mathbb{R}^n$, the Jacobian matrix $J_f(x)$ of $f$ is orthogonal.
What is the necessary and sufficient condition on $f$ for proposition $(\mathcal{P})$ to hold?

Let $f$ be a function of class $C^2$ from $\mathbb{R}^n$ to itself. We consider the proposition $(\mathcal{P})$: for all $x$ in $\mathbb{R}^n$, the Jacobian matrix $J_f(x)$ of $f$ is orthogonal.
If $g$ is a function of class $C^2$ from $\mathbb{R}^n$ to $\mathbb{R}$, we denote $\Delta_g(x) = \sum_{i=1}^n \frac{\partial^2 g}{\partial x_i^2}(x)$ (Laplacian of $g$ at $x$). Show that $(\mathcal{P})$ is equivalent to the proposition $$\text{For every function } g \text{ of class } C^2 \text{ from } \mathbb{R}^n \text{ to } \mathbb{R},\quad \Delta_{g \circ f} = (\Delta_g) \circ f.$$

Let $q \in \mathcal { Q } ( V )$ be a non-degenerate quadratic form on $V$.
(a) Let $V ^ { \prime }$ be a $\mathbb { K }$-vector space of finite dimension and $q ^ { \prime }$ a quadratic form on $V ^ { \prime }$. Prove that if $q$ and $q ^ { \prime }$ are isometric, then $q ^ { \prime }$ is in $\mathcal { Q } \left( V ^ { \prime } \right)$, that is, non-degenerate.
(b) For $x \neq 0$, we denote $\{ x \} ^ { \perp } : = \{ y \in V \mid \widetilde { q } ( x , y ) = 0 \}$. Show that $\{ x \} ^ { \perp }$ is a vector subspace of $V$ of dimension $n - 1$.
(c) Under what condition on $x$ is the subspace $\{ x \} ^ { \perp }$ a complement of the line $\mathbb { K } x$ in $V$ ?

We denote by $K$ the function defined from $[0,1]^2$ to $\mathbb{R}$ by the following relation: $K(s,t) = (1-s)t$ if $0 \leq t \leq s \leq 1$ and $K(s,t) = (1-t)s$ otherwise. We denote by $T$ the application defined on $E = C([0,1], \mathbb{R})$, equipped with the norm $\|.\|_2 = \sqrt{\int_0^1 |f(x)|^2 \, dx}$, by the relation: $$\forall f \in E, \quad \forall s \in [0,1], \quad T(f)(s) = \int_0^1 K(s,t) f(t) \, dt$$ Show that $T \in \mathcal{L}(E)$.

We denote by $K$ the function defined from $[0,1]^2$ to $\mathbb{R}$ by the following relation: $K(s,t) = (1-s)t$ if $0 \leq t \leq s \leq 1$ and $K(s,t) = (1-t)s$ otherwise. We denote by $T$ the application defined on $E = C([0,1], \mathbb{R})$, equipped with the norm $\|.\|_2 = \sqrt{\int_0^1 |f(x)|^2 \, dx}$, by the relation: $$\forall f \in E, \quad \forall s \in [0,1], \quad T(f)(s) = \int_0^1 K(s,t) f(t) \, dt$$ Show that $T$ is injective.

We call an H-system of endomorphisms of $E$ any finite family of symmetries of $E$ that anticommute pairwise, that is, any finite family $(S_1, \ldots, S_p)$ of endomorphisms of $E$ such that $$\left\{ \begin{array}{lrl} \forall i & S_i \circ S_i & = \operatorname{Id}_E \\ \forall i \neq j & S_i \circ S_j + S_j \circ S_i & = 0 \end{array} \right.$$ Similarly, we call an H-system of matrices of size $n$ any finite family $(A_1, \ldots, A_p)$ of matrices of $\mathcal{M}_n(\mathbb{C})$ such that $$\left\{ \begin{aligned} \forall i & A_i^2 & = I_n \\ \forall i \neq j & A_i A_j + A_j A_i & = 0 \end{aligned} \right.$$ In both cases, $p$ is called the length of the H-system. Show that the length $p$ of an H-system of endomorphisms of $E$ is bounded above by $n^2$.

We call an H-system of endomorphisms of $E$ any finite family of symmetries of $E$ that anticommute pairwise, that is, any finite family $(S_1, \ldots, S_p)$ of endomorphisms of $E$ such that $$\left\{ \begin{array}{lrl} \forall i & S_i \circ S_i & = \operatorname{Id}_E \\ \forall i \neq j & S_i \circ S_j + S_j \circ S_i & = 0 \end{array} \right.$$ Similarly, we call an H-system of matrices of size $n$ any finite family $(A_1, \ldots, A_p)$ of matrices of $\mathcal{M}_n(\mathbb{C})$ such that $$\left\{ \begin{aligned} \forall i & A_i^2 & = I_n \\ \forall i \neq j & A_i A_j + A_j A_i & = 0 \end{aligned} \right.$$ In both cases, $p$ is called the length of the H-system. Show that the existence of an H-system $(S_1, \ldots, S_p)$ of $E$ is equivalent to the existence of an H-system of matrices of size $n$. Deduce that the length of an H-system of $E$ depends only on the dimension $n$ of $E$ and not on the space $E$.

We call an H-system of endomorphisms of $E$ any finite family of symmetries of $E$ that anticommute pairwise, that is, any finite family $(S_1, \ldots, S_p)$ of endomorphisms of $E$ such that $$\left\{ \begin{array}{lrl} \forall i & S_i \circ S_i & = \operatorname{Id}_E \\ \forall i \neq j & S_i \circ S_j + S_j \circ S_i & = 0 \end{array} \right.$$ We denote by $p(n)$ the largest integer $p \geqslant 1$ such that $E$ admits an H-system of cardinality $p$. Let $n$ be an odd integer. Prove that $p(n) = 1$.

We denote by $p(n)$ the largest integer $p \geqslant 1$ such that $E$ admits an H-system of cardinality $p$. We assume here that $n$ is even and we set $n = 2m$. We consider:

• an H-system $(S_1, \ldots, S_p, T, U)$ of $E$,
• the subspace $E_0 = F_T = \operatorname{Ker}(T - \mathrm{Id})$,
• for $j \in \llbracket 1, p \rrbracket$, the endomorphism $R_j = \mathrm{i} U \circ S_j$ of $E$.

a) Show that, for all $j \in \llbracket 1, p \rrbracket$, the subspace $E_0$ is stable under $R_j$. b) For $j \in \llbracket 1, p \rrbracket$, let $s_j$ be the endomorphism of $E_0$ induced by $R_j$. Show that $(s_1, \ldots, s_p)$ is an H-system of $E_0$. c) Deduce that $p(2m) \leqslant p(m) + 2$.

We denote by $p(n)$ the largest integer $p \geqslant 1$ such that $E$ admits an H-system of cardinality $p$. Show that if $n = 2^d m$ with $m$ odd, then $p(n) \leqslant 2d + 1$.

We denote by $p(n)$ the largest integer $p \geqslant 1$ such that $E$ admits an H-system of cardinality $p$. Let $N = p(n)$ and $(a_1, \ldots, a_N)$ be an H-system of matrices of size $n$, that is, such that $$\forall i, a_i^2 = I_n \quad \text{and} \quad \forall i \neq j, a_i a_j + a_j a_i = 0$$ By considering the following matrices of $\mathcal{M}_{2n}(\mathbb{C})$ written in block form $$A_j = \left( \begin{array}{cc} a_j & 0 \\ 0 & -a_j \end{array} \right) (j \in \llbracket 1, N \rrbracket), \quad A_{N+1} = \left( \begin{array}{cc} 0 & I_n \\ I_n & 0 \end{array} \right), \quad A_{N+2} = \left( \begin{array}{cc} 0 & \mathrm{i} I_n \\ -\mathrm{i} I_n & 0 \end{array} \right)$$ show that $p(2n) \geqslant N + 2$.

We denote by $p(n)$ the largest integer $p \geqslant 1$ such that $E$ admits an H-system of cardinality $p$. Determine $p(n)$ as a function of the unique integer $d \in \mathbb{N}$ such that $n$ can be written as $n = 2^d m$ with $m$ odd.

We denote by $p(n)$ the largest integer $p \geqslant 1$ such that $E$ admits an H-system of cardinality $p$. Write, for each of the integers $n = 1, 2, 4$, an H-system of matrices of size $n$ of length $p(n)$.

Let $n \geqslant 1$ be a natural number. We equip $\mathbb{R}^n$ with the usual inner product and the usual Euclidean norm defined, for all $X = (x_1, \ldots, x_n)$ and $Y = (y_1, \ldots, y_n)$ of $\mathbb{R}^n$, by $$(X \mid Y) = \sum_{i=1}^n x_i y_i \quad \text{and} \quad \|X\| = \sqrt{\sum_{k=1}^n x_k^2}$$ We study the existence of a bilinear map $B_n: (\mathbb{R}^n)^2 \rightarrow \mathbb{R}^n$ satisfying $$\forall X, Y \in \mathbb{R}^n, \quad \|B_n(X, Y)\| = \|X\| \times \|Y\|$$ Show the existence of such a bilinear map $B_n$ when $n$ is one of the integers $1, 2, 4$.
For $n = 2$ (respectively 4) one may consider the product of two complex numbers (respectively of two quaternions).

Let $n \geqslant 1$ be a natural number. We equip $\mathbb{R}^n$ with the usual inner product and the usual Euclidean norm defined, for all $X = (x_1, \ldots, x_n)$ and $Y = (y_1, \ldots, y_n)$ of $\mathbb{R}^n$, by $$(X \mid Y) = \sum_{i=1}^n x_i y_i \quad \text{and} \quad \|X\| = \sqrt{\sum_{k=1}^n x_k^2}$$ We study the existence of a bilinear map $B_n: (\mathbb{R}^n)^2 \rightarrow \mathbb{R}^n$ satisfying $$\forall X, Y \in \mathbb{R}^n, \quad \|B_n(X, Y)\| = \|X\| \times \|Y\|$$ Using question II.B.2 show, for $n = 8$, the existence of a bilinear map satisfying the above. We do not ask you to explicitly write down a bilinear map $B_8$, but only to prove its existence.

We assume that $n \geqslant 3$ and that there exists a bilinear map $B$ such that $$\forall X, Y \in \mathbb{R}^n, \quad \|X\| \times \|Y\| = \|B(X, Y)\|$$ Let $(e_1, \ldots, e_n)$ be the canonical basis of $\mathbb{R}^n$ and, for $i \in \llbracket 1, n \rrbracket$, let $u_i$ be the endomorphism of $\mathbb{R}^n$ defined by $$\forall X \in \mathbb{R}^n, \quad u_i(X) = B(X, e_i)$$ The matrix of $u_i$ in the canonical basis of $\mathbb{R}^n$ will be denoted $A_i$. a) Prove that, for all $X \in \mathbb{R}^n$, we have $$\forall Y = (y_1, \ldots, y_n) \in \mathbb{R}^n, \quad \sum_{i,j=1}^n y_i y_j (u_i(X) \mid u_j(X)) = \|X\|^2 \sum_{i=1}^n y_i^2$$ b) Deduce that the endomorphisms $u_i$ satisfy the relations $$\forall i, j = 1, \ldots, n, \forall X \in \mathbb{R}^n, \quad \|u_i(X)\| = \|X\| \text{ and } i \neq j \Rightarrow (u_i(X) \mid u_j(X)) = 0$$ and more generally $$\forall i, j = 1, \ldots, n, \forall X, X' \in \mathbb{R}^n, \quad (u_i(X) \mid u_i(X')) = (X \mid X') \text{ and } i \neq j \Rightarrow (u_i(X) \mid u_j(X')) + (u_j(X) \mid u_i(X')) = 0$$ c) Prove that the matrices $A_i$ satisfy the relations $\forall i, j = 1, \ldots, n, \quad {}^t A_i A_i = I_n$ and $i \neq j \Rightarrow {}^t A_i A_j + {}^t A_j A_i = 0$.

We assume that $n \geqslant 3$ and that there exists a bilinear map $B$ such that $$\forall X, Y \in \mathbb{R}^n, \quad \|X\| \times \|Y\| = \|B(X, Y)\|$$ Let $(e_1, \ldots, e_n)$ be the canonical basis of $\mathbb{R}^n$ and, for $i \in \llbracket 1, n \rrbracket$, let $u_i$ be the endomorphism of $\mathbb{R}^n$ defined by $\forall X \in \mathbb{R}^n, u_i(X) = B(X, e_i)$. The matrix of $u_i$ in the canonical basis of $\mathbb{R}^n$ will be denoted $A_i$. We denote by $p(n)$ the largest integer $p \geqslant 1$ such that $E$ admits an H-system of cardinality $p$. For $j = 1, \ldots, n-1$ we denote by $S_j$ the complex matrix $S_j = \mathrm{i} {}^t A_n A_j$. a) Prove that $(S_1, \ldots, S_{n-1})$ is an H-system. b) Deduce that we have the inequality $p(n) \geqslant n - 1$ where $p(n)$ is defined in section I.C.

We assume that $n \geqslant 3$ and that there exists a bilinear map $B$ such that $$\forall X, Y \in \mathbb{R}^n, \quad \|X\| \times \|Y\| = \|B(X, Y)\|$$ We denote by $p(n)$ the largest integer $p \geqslant 1$ such that $E$ admits an H-system of cardinality $p$. Prove that $n$ is an element of $\{1, 2, 4, 8\}$.

In this part, $n$ is a non-zero natural integer, $M$ is in $\mathcal{M}_n(\mathbb{R})$ and $f$ is the endomorphism of $E = \mathcal{M}_{n,1}(\mathbb{R})$ defined by $f(X) = MX$ for all $X$ in $E$.
If we denote $X_i = \begin{pmatrix} \delta_{1,i} \\ \vdots \\ \delta_{n,i} \end{pmatrix}$ where $\delta_{k,l} = \begin{cases} 1 & \text{if } k = l \\ 0 & \text{if } k \neq l \end{cases}$ and $\mathcal{B}_n = (X_i)_{1 \leqslant i \leqslant n}$ the canonical basis of $E$, what is the matrix of $f$ in $\mathcal{B}_n$?