For $(A, B) \in \mathcal{M}_d(\mathbb{R})^2$ such that $A$ and $B$ commute, show that $\exp(\mathrm{i}A)\exp(\mathrm{i}B) = \exp(\mathrm{i}(A+B))$.

For every $A \in \mathcal{M}_d(\mathbb{R})$, we define $$\cos(A) = \sum_{n=0}^{+\infty} (-1)^n \frac{A^{2n}}{(2n)!} \quad \text{and} \quad \sin(A) = \sum_{n=0}^{+\infty} (-1)^n \frac{A^{2n+1}}{(2n+1)!}$$ Show $$\forall A \in \mathcal{M}_d(\mathbb{R}), \quad \cos(A)^2 + \sin(A)^2 = I_d$$

We fix a matrix $A \in \mathcal{M}_d(\mathbb{R})$.
For $R$ large enough, show that, for every $\theta \in \mathbb{R}$, the matrix $(R\mathrm{e}^{\mathrm{i}\theta} I_d - A)$ is invertible in $\mathcal{M}_d(\mathbb{C})$, and that its inverse is the matrix $$\left(R\mathrm{e}^{\mathrm{i}\theta}\right)^{-1} \sum_{n=0}^{+\infty} \left(R\mathrm{e}^{\mathrm{i}\theta}\right)^{-n} A^n$$

We are given a function $\xi : \mathbb{R} \rightarrow \mathbb{R}$ and we define a function $f_\xi : \mathcal{M}_d(\mathbb{R}) \rightarrow \mathcal{M}_d(\mathbb{R})$ such that $$\forall A \in \mathcal{M}_d(\mathbb{R}), \quad f_\xi(A) = \left(\xi\left(A_{i,j}\right)\right)_{1 \leqslant i,j \leqslant d}$$ We propose to determine the continuous functions $\xi : \mathbb{R} \rightarrow \mathbb{R}$ such that $$\forall A \in \mathcal{M}_d(\mathbb{R}), \quad A \text{ invertible} \Rightarrow f_\xi(A) \text{ invertible} \tag{V.1}$$
Determine the continuous functions $\xi$ satisfying condition (V.1) when $d = 1$.

We are given a continuous function $\xi : \mathbb{R} \rightarrow \mathbb{R}$ satisfying condition (V.1) (with $d \geqslant 2$), where $$\forall A \in \mathcal{M}_d(\mathbb{R}), \quad A \text{ invertible} \Rightarrow f_\xi(A) = \left(\xi(A_{i,j})\right)_{1\leqslant i,j\leqslant d} \text{ invertible} \tag{V.1}$$
Show $$\forall (a, b, c, d) \in \mathbb{R}^4, \quad ad \neq bc \Rightarrow \xi(a)\xi(d) \neq \xi(b)\xi(c)$$ One may consider the matrix $\begin{pmatrix} a & b & 0 & \cdots & 0 \\ c & d & 0 & \cdots & 0 \\ c & d & & & \\ \vdots & \vdots & & I_{d-2} & \\ c & d & & & \end{pmatrix}$.

We are given a continuous function $\xi : \mathbb{R} \rightarrow \mathbb{R}$ satisfying condition (V.1) (with $d \geqslant 2$), where $$\forall A \in \mathcal{M}_d(\mathbb{R}), \quad A \text{ invertible} \Rightarrow f_\xi(A) = \left(\xi(A_{i,j})\right)_{1\leqslant i,j\leqslant d} \text{ invertible} \tag{V.1}$$
Deduce that the function $\xi$ is injective, then that it is strictly monotone on $\mathbb{R}$.

We are given a continuous function $\xi : \mathbb{R} \rightarrow \mathbb{R}$ satisfying condition (V.1) (with $d \geqslant 2$), where $$\forall A \in \mathcal{M}_d(\mathbb{R}), \quad A \text{ invertible} \Rightarrow f_\xi(A) = \left(\xi(A_{i,j})\right)_{1\leqslant i,j\leqslant d} \text{ invertible} \tag{V.1}$$
Show that the function $\xi$ does not vanish on $\mathbb{R}^*$.

We are given a continuous function $\xi : \mathbb{R} \rightarrow \mathbb{R}$ satisfying condition (V.1) (with $d \geqslant 2$), where $$\forall A \in \mathcal{M}_d(\mathbb{R}), \quad A \text{ invertible} \Rightarrow f_\xi(A) = \left(\xi(A_{i,j})\right)_{1\leqslant i,j\leqslant d} \text{ invertible} \tag{V.1}$$
The purpose of this question is to show $\xi(0) = 0$.
1) Show that if $\xi(0) \neq 0$, then there exists $\alpha > 0$ such that $\xi(0)\xi(2) = \xi(1)\xi(\alpha)$.
2) Conclude.

For $\lambda \in \mathbb{R}$, calculate the determinant of the matrix $A_\lambda \in \mathcal{M}_d(\mathbb{R})$ having only 1's off the diagonal and only $\lambda$ on the diagonal.

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 $B^3 = A$.

Let $F$ and $G$ be two supplementary subspaces of $E$ and $s$ the symmetry with respect to $F$ parallel to $G$. a) Show that $F = F_s$ and $G = G_s$. b) Show that $s \circ s = \operatorname{Id}_E$. Deduce that $s$ is an automorphism of $E$. c) Determine the eigenvalues and eigenspaces of $s$. We will discuss according to the subspaces $F$ and $G$.

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. Show that $(\mathbb{H} \backslash \{0\}, \times)$ is a non-commutative subgroup of the linear group $(\mathrm{GL}_2(\mathbb{C}), \times)$.

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 $(N(a) + N(b))(N(c) + N(d)) = N(ac - d^* b) + N(bc^* + da)$.

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.

Let $A$ and $B$ be two matrices of $\mathcal { M } _ { n } ( \mathbb { R } )$. Show that if, for all $X$ and $Y$ of $\mathbb { R } ^ { n } , { } ^ { t } X A Y = { } ^ { t } X B Y$ then $A = B$.

We define $$\varphi _ { p + 1 } \left( v , v ^ { \prime } \right) = { } ^ { t } V \Delta _ { p + 1 } V ^ { \prime } = v _ { 1 } v _ { 1 } ^ { \prime } - \sum _ { i = 2 } ^ { p + 1 } v _ { i } v _ { i } ^ { \prime }$$ and $$q _ { p + 1 } ( v ) = \varphi _ { p + 1 } ( v , v )$$ Express $\varphi _ { p + 1 } \left( v , v ^ { \prime } \right)$ as a function of $q _ { p + 1 } \left( v + v ^ { \prime } \right)$ and $q _ { p + 1 } \left( v - v ^ { \prime } \right)$.