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 $$\text{For every function } g \text{ of class } C^2 \text{ from } \mathbb{R}^n \text{ to } \mathbb{R},\quad \Delta_{g \circ f} = (\Delta_g) \circ f.$$

We denote by $\mathrm{GL}_2(\mathbb{Z})$ the set of invertible elements of the ring $\mathcal{M}_2(\mathbb{Z})$, equipped with its usual addition and multiplication.
Justify that an element $M$ of $\mathcal{M}_2(\mathbb{Z})$ belongs to $\mathrm{GL}_2(\mathbb{Z})$ if and only if $|\det M| = 1$.

We denote by $\mathrm{GL}_2(\mathbb{Z})$ the set of invertible elements of $\mathcal{M}_2(\mathbb{Z})$. The Dickson polynomials $(D_n)_{n \in \mathbb{N}}$ and $(E_n)_{n \in \mathbb{N}}$ satisfy the recurrences $D_{n+2}(x,a) = x D_{n+1}(x,a) - a D_n(x,a)$ and $E_{n+2}(x,a) = x E_{n+1}(x,a) - a E_n(x,a)$ with $D_0 = 2, D_1(x,a) = x, E_0 = 1, E_1(x,a) = x$.
Let $B \in \mathrm{GL}_2(\mathbb{Z})$. We denote, in this question only, $\sigma = \operatorname{Tr} B$ and $\nu = \det B$. Show for every $n \geqslant 2$, the equality $$B^n = E_{n-1}(\sigma, \nu) \cdot B - \nu E_{n-2}(\sigma, \nu) \cdot I_2$$ where $I_2$ is the identity matrix of order 2.
Establish that $\operatorname{Tr}(B^n) = D_n(\sigma, \nu)$.

We denote by $\mathrm{GL}_2(\mathbb{Z})$ the set of invertible elements of $\mathcal{M}_2(\mathbb{Z})$. For $B \in \mathrm{GL}_2(\mathbb{Z})$ with $\sigma = \operatorname{Tr} B$ and $\nu = \det B$, we have $B^n = E_{n-1}(\sigma,\nu) \cdot B - \nu E_{n-2}(\sigma,\nu) \cdot I_2$ and $\operatorname{Tr}(B^n) = D_n(\sigma,\nu)$.
We denote $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$, $\tau = \operatorname{Tr} A$, $\delta = \det A$.
Deduce that if $A$ is an $n$-th power ($n \geqslant 2$) in $\mathrm{GL}_2(\mathbb{Z})$, then there exist $\sigma \in \mathbb{Z}$ and $\nu \in \{-1,1\}$ such that
i. $E_{n-1}(\sigma, \nu)$ divides $b$, $c$ and $a - d$. One will briefly justify that $E_{n-1}(\sigma, \nu)$ is indeed an integer.
ii. $\tau = D_n(\sigma, \nu)$ and $\delta = \nu^n$.

We denote by $\mathrm{GL}_2(\mathbb{Z})$ the set of invertible elements of $\mathcal{M}_2(\mathbb{Z})$. We denote $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$, $\tau = \operatorname{Tr} A$, $\delta = \det A$.
Let $A$ be an element of $\mathrm{GL}_2(\mathbb{Z})$ for which there exist $\sigma \in \mathbb{Z}$ and $\nu \in \{-1,1\}$ satisfying: i. $E_{n-1}(\sigma, \nu)$ divides $b$, $c$ and $a - d$. ii. $\tau = D_n(\sigma, \nu)$ and $\delta = \nu^n$.
For simplicity, we denote $p = E_{n-1}(\sigma, \nu)$. We then define a matrix $B = \begin{pmatrix} r & s \\ t & u \end{pmatrix}$ with $$r = \frac{1}{2}\left(\sigma + \frac{a-d}{p}\right) \quad s = \frac{b}{p} \quad t = \frac{c}{p} \quad u = \frac{1}{2}\left(\sigma - \frac{a-d}{p}\right)$$
a) By introducing a complex root of the polynomial $X^2 - \sigma X + \nu$ and using the relation $D_n(x + a/x, a) = x^n + a^n/x^n$, show that $$\tau^2 - 4\delta = p^2(\sigma^2 - 4\nu) \quad \text{then} \quad ru - st = \nu$$ Deduce that $B$ belongs to $\mathrm{GL}_2(\mathbb{Z})$.
b) Show that $A = B^n$.

We denote by $\mathrm{GL}_2(\mathbb{Z})$ the set of invertible elements of $\mathcal{M}_2(\mathbb{Z})$. The necessary and sufficient condition for $A \in \mathrm{GL}_2(\mathbb{Z})$ to be an $n$-th power is the existence of $\sigma \in \mathbb{Z}$ and $\nu \in \{-1,1\}$ such that $E_{n-1}(\sigma,\nu)$ divides $b$, $c$ and $a-d$, and $\tau = D_n(\sigma,\nu)$, $\delta = \nu^n$.
Show that the matrix $A = \begin{pmatrix} 7 & 10 \\ 5 & 7 \end{pmatrix}$ is a cube in $\mathrm{GL}_2(\mathbb{Z})$ and determine a matrix $B \in \mathrm{GL}_2(\mathbb{Z})$ such that $B^3 = A$.

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 $p(n)$.

For $(a, b) \in \mathbb{C}^2$, we denote by $M(a, b)$ the square complex matrix $M(a, b) = \left( \begin{array}{cc} a & -b \\ \bar{b} & \bar{a} \end{array} \right) \in \mathcal{M}_2(\mathbb{C})$. A matrix of the form $M(a, b)$ will be called a quaternion. We will consider in particular the quaternions $e = I_2 = M(1, 0)$, $I = M(0, 1)$, $J = M(\mathrm{i}, 0)$, $K = M(0, -\mathrm{i})$ and we will denote by $\mathbb{H} = \{M(a, b) \mid (a, b) \in \mathbb{C}^2\}$ the subset of $\mathcal{M}_2(\mathbb{C})$ consisting of all quaternions. We equip the set $\mathcal{C} = \mathcal{M}_2(\mathbb{C})$ of complex matrices with two rows and two columns with addition $+$, multiplication $\times$ in the usual sense, and multiplication by a real number denoted $\cdot$. a) Give, without justification, a basis and the dimension of $\mathcal{C}$ over the field $\mathbb{R}$. b) Show that $\mathbb{H}$ is a real vector subspace of $\mathcal{C}$ and that $\{e, I, J, K\}$ is a basis for it over the field $\mathbb{R}$. c) Show that $\mathbb{H}$ is stable under multiplication.

For $(a, b) \in \mathbb{C}^2$, we denote by $M(a, b)$ the square complex matrix $M(a, b) = \left( \begin{array}{cc} a & -b \\ \bar{b} & \bar{a} \end{array} \right) \in \mathcal{M}_2(\mathbb{C})$. A matrix of the form $M(a, b)$ will be called a quaternion. We will consider in particular the quaternions $e = I_2 = M(1, 0)$, $I = M(0, 1)$, $J = M(\mathrm{i}, 0)$, $K = M(0, -\mathrm{i})$ and we will denote by $\mathbb{H} = \{M(a, b) \mid (a, b) \in \mathbb{C}^2\}$ the subset of $\mathcal{M}_2(\mathbb{C})$ consisting of all quaternions. a) Calculate the products pairwise of the matrices $e, I, J, K$. Present the results in a double-entry table. b) Deduce that $(\mathrm{i} I, \mathrm{i} J, \mathrm{i} K)$ is an H-system.

For $(a, b) \in \mathbb{C}^2$, we denote by $M(a, b)$ the square complex matrix $M(a, b) = \left( \begin{array}{cc} a & -b \\ \bar{b} & \bar{a} \end{array} \right) \in \mathcal{M}_2(\mathbb{C})$. Every element $q \in \mathbb{H}$ can be written uniquely as $q = xe + yI + zJ + tK$ with $x, y, z, t \in \mathbb{R}$. For $x, y, z, t \in \mathbb{R}$ and $q = xe + yI + zJ + tK \in \mathbb{H}$ we set $q^* = xe - yI - zJ - tK \in \mathbb{H}$ and $N(q) = x^2 + y^2 + z^2 + t^2 \in \mathbb{R}_+$. a) Verify that, for all $q \in \mathbb{H}$, $q^*$ is the transpose of the matrix whose coefficients are the conjugates of the coefficients of $q$. b) Deduce that, for all $(q, r) \in \mathbb{H}^2$, $(qr)^* = r^* q^*$. c) Show that $q^{**} = q$ for all $q \in \mathbb{H}$ and that $q \mapsto q^*$ is an automorphism of the $\mathbb{R}$-vector space $\mathbb{H}$. d) For $q \in \mathbb{H}$, express $qq^*$ in terms of $N(q)$. Deduce the relation valid for all $(q, r) \in \mathbb{H}^2$ $$N(qr) = N(q)N(r)$$

For $(a, b) \in \mathbb{C}^2$, we denote by $M(a, b)$ the square complex matrix $M(a, b) = \left( \begin{array}{cc} a & -b \\ \bar{b} & \bar{a} \end{array} \right) \in \mathcal{M}_2(\mathbb{C})$. Every element $q \in \mathbb{H}$ can be written uniquely as $q = xe + yI + zJ + tK$ with $x, y, z, t \in \mathbb{R}$. For $x, y, z, t \in \mathbb{R}$ and $q = xe + yI + zJ + tK \in \mathbb{H}$ we set $q^* = xe - yI - zJ - tK \in \mathbb{H}$ and $N(q) = x^2 + y^2 + z^2 + t^2 \in \mathbb{R}_+$. a) Let $(x, y, z, t) \in \mathbb{R}^4$ and $q = xe + yI + zJ + tK$. Express the trace of the matrix $q \in \mathcal{M}_2(\mathbb{C})$ in terms of the real number $x$. b) Deduce that, for all $(q, r) \in \mathbb{H}^2$, $qr - rq = q^* r^* - r^* q^*$. c) Let $a, b, c, d$ be quaternions. Establish the relation $(acb^*)d + d^*(acb^*)^* = (acb^*)^* d^* + d(acb^*)$.
Deduce the identity $(N(a) + N(b))(N(c) + N(d)) = N(ac - d^* b) + N(bc^* + da)$.

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 $\{1, 2, 4, 8\}$.

Let $(A, +, \times)$ be a commutative ring. For $p \in \mathbb{N}^*$, we denote by $C_p(A)$ the set of sums of $p$ squares of elements of $A$. Prove that for every ring $A$, the sets $C_p(A)$ are stable under multiplication when $p$ equals $1, 2, 4$ or $8$.
You may use the bilinear forms $B_p$ defined in part III and, if necessary, restrict yourself to the case where the ring $A$ is the ring $\mathbb{Z}$ of integers.