Let $D \in M_p(\mathbb{K})$ be a diagonal matrix. Let $(\Delta, +)$ be the additive subgroup of $M_p(\mathbb{R})$ formed by diagonal matrices.
Show that $E$ defines a group morphism from $(\Delta, +)$ to $(GL_p(\mathbb{R}), \times)$.

Show that $A \in \mathrm{SO}(2)$ if and only if there exists a real $t$ such that $A = R_t$ with $R_t = \left(\begin{array}{rr} \cos t & -\sin t \\ \sin t & \cos t \end{array}\right)$.

Write a procedure or function in Maple or Mathematica that takes as input a quadruple $(a,b,c,d)$ of reals and returns, when possible, a real $t$ such that $\left(\begin{array}{ll} a & b \\ c & d \end{array}\right) = R_t$ and an error message otherwise.

Verify that the map which associates to every real $t$ the matrix $R_t$ is a surjective homomorphism from the group $(\mathbb{R},+)$ onto the group $(\mathrm{SO}(2),\times)$. Is this homomorphism bijective?

Show that, for all $t$ in $\mathbb{R}$ and all non-zero $u$ in $\mathbb{R}^2$, $t$ is a measure of the oriented angle $(u\widehat{\rho_t(u)})$, where $\rho_t$ is the endomorphism (the rotation of angle $t$) $f_{R_t}$ canonically associated with $R_t$.

Show that for every matrix $A$ of $\mathrm{O}(2)$ such that $\det(A) = -1$, there exists a real $t$ such that $$A = \left(\begin{array}{cr} \cos(2t) & \sin(2t) \\ \sin(2t) & -\cos(2t) \end{array}\right)$$

Show that for all $A$ in $\mathcal{M}_n(\mathbb{R})$ we have $A$ dos $A$, that for all $(A,B)$ in $\mathcal{M}_n(\mathbb{R})^2$ if $A$ dos $B$ then $B$ dos $A$, and that for all $(A,B,C)$ in $\mathcal{M}_n(\mathbb{R})^3$ if $A$ dos $B$ and $B$ dos $C$ then $A$ dos $C$.

What are the matrices directly orthogonally similar to $\alpha I_n$ for $\alpha$ real?

What are the matrices directly orthogonally similar to $A$ if $A$ belongs to $\mathrm{SO}(2)$?

What are the matrices directly orthogonally similar to $K_2 = \left(\begin{array}{rr} 1 & 0 \\ 0 & -1 \end{array}\right)$?

Show that $\left(\begin{array}{ll} 0 & 0 \\ 0 & 2 \end{array}\right)$ and $\left(\begin{array}{ll} 1 & 1 \\ 1 & 1 \end{array}\right)$ are directly orthogonally similar.

Show that $\left(\begin{array}{ll} 1 & 0 \\ 0 & 2 \end{array}\right)$ and $\left(\begin{array}{rr} 3 & 2 \\ -1 & 0 \end{array}\right)$ are similar but are not orthogonally similar.

Show that $\left(\begin{array}{rr} 3 & 2 \\ -1 & 0 \end{array}\right)$ and its transpose are orthogonally similar but are not directly orthogonally similar.

We assume that the conditions of questions 4 and 5 are satisfied and that $\lambda(0) = 0, \mu(0) = 1$.
7a. Show that $F \in \mathrm{GL}(V)$.
7b. Show that $E$ and $F$ are not of finite order in the group $\mathrm{GL}(V)$.
7c. Calculate the kernel of $H$ and show that $H^r \neq \operatorname{Id}_V$ for $r \geq 1$.

We assume that the conditions of questions 4 and 5 are satisfied and that $\lambda(0) = 0, \mu(0) = 1$. We denote by $\mathbf{C}[X]$ the algebra of polynomials with complex coefficients in one indeterminate $X$.
8a. Show that $\mathbf{C}[E]$ is isomorphic (as an algebra) to $\mathbf{C}[X]$.
8b. Show that $\mathbf{C}[F]$ is isomorphic (as an algebra) to $\mathbf{C}[X]$.
8c. Show that $\mathbf{C}[H]$ is isomorphic (as an algebra) to $\mathbf{C}[X]$.

Let $W_{\ell} = \bigoplus_{0 \leq i < \ell} \mathbf{C} v_i$ and $a \in \mathbf{C}^*$. We consider the element $G_a$ of $\mathcal{L}(W_{\ell})$ whose matrix in the basis $\{v_i\}_{0 \leq i < \ell}$ is: $$\left(\begin{array}{cccccc} 0 & 0 & 0 & \cdots & 0 & a \\ 1 & 0 & 0 & \cdots & 0 & 0 \\ 0 & 1 & 0 & \cdots & 0 & 0 \\ \vdots & 0 & \ddots & \ddots & \vdots & \vdots \\ 0 & \vdots & \ddots & \ddots & 0 & 0 \\ 0 & 0 & \ldots & 0 & 1 & 0 \end{array}\right)$$
10a. Calculate $G_a^{\ell}$. Show that $G_a$ is diagonalizable.
10b. Let $b$ be an $\ell$-th root of $a$. Calculate the eigenvectors of $G_a$ and the associated eigenvalues in terms of $b, q$ and the $v_i$.

Let $W_{\ell} = \bigoplus_{0 \leq i < \ell} \mathbf{C} v_i$ and $a \in \mathbf{C}^*$. We define a linear map $P_a : V \rightarrow V$ by $P_a(v_i) = a^p v_r$ where for $i \in \mathbf{Z}$, we define $r$ and $p$ respectively as the remainder and quotient of the Euclidean division of $i$ by $\ell$; in other words, $i = p\ell + r$ where $0 \leq r < \ell$ and $p \in \mathbf{Z}$. Show that $P_a$ is a projector with image $W_{\ell}$.

Let $\lambda, \mu \in \mathbf{C}^{\mathbf{Z}}$ and $H \in \mathcal{L}(V)$ defined by $H(v_i) = \lambda(i) v_i$. Show that $H \circ E = q^2 E \circ H$ if and only if for all $i \in \mathbf{Z}, \lambda(i) = \lambda(0) q^{-2i}$.

We equip $\mathbb{C}[X]$ with the internal composition law given by composition, denoted $\circ$. We seek families $(F_n)_{n \in \mathbb{N}}$ of complex polynomials satisfying $$\forall n \in \mathbb{N}, \quad \deg F_n = n \quad \text{and} \quad \forall (m,n) \in \mathbb{N}^2, \quad F_n \circ F_m = F_m \circ F_n \tag{III.1}$$
Show that the family $(T_n)_{n \in \mathbb{N}}$ satisfies property (III.1). One may compare $T_n \circ T_m$ and $T_{mn}$.

We equip $\mathbb{C}[X]$ with the internal composition law given by composition, denoted $\circ$. We denote by $G$ the set of complex polynomials of degree 1.
Verify that $G$ is a group for the law $\circ$.

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 $(\mathrm{GL}_2(\mathbb{C}), \times)$.

We denote $\mathbb{G} = \{xe + yI + zJ + tK \mid x, y, z, t \in \mathbb{Z}\}$ the set of ``integer'' quaternions. For $q \in \mathbb{H}$, $N(q) = x^2 + y^2 + z^2 + t^2$ where $q = xe + yI + zJ + tK$. a) Show that $\mathbb{G}$ is a subgroup of $\mathbb{H}$ for addition and that it is stable under multiplication. b) Show that for every $q \in \mathbb{H}$, there exists $\mu \in \mathbb{G}$ such that $N(q - \mu) \leqslant 1$. c) What is the set of $q \in \mathbb{H}$ such that $\forall \mu \in \mathbb{G}, N(q - \mu) \geqslant 1$?

Show that $O ( 1 , p ) = O ^ { + } ( 1 , p ) \cup O ^ { - } ( 1 , p )$.

Show that the set $O ( 1 , p )$ is a subgroup of $G L _ { p + 1 } ( \mathbb { R } )$ and that $O ^ { + } ( 1 , p )$ is a subgroup of $O ( 1 , p )$.

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 $$q _ { p + 1 } ( v ) = \varphi _ { p + 1 } ( v , v )$$ Let $L \in \mathcal { M } _ { p + 1 } ( \mathbb { R } )$ and $f$ the endomorphism of $\mathbb { R } ^ { p + 1 }$ canonically associated.
Show that the following three assertions are equivalent:
i. $L \in O ( 1 , p )$;
ii. $\forall \left( v , v ^ { \prime } \right) \in \left( \mathbb { R } ^ { p + 1 } \right) ^ { 2 } , \varphi _ { p + 1 } \left( f ( v ) , f \left( v ^ { \prime } \right) \right) = \varphi _ { p + 1 } \left( v , v ^ { \prime } \right)$;
iii. $\forall v \in \mathbb { R } ^ { p + 1 } , q _ { p + 1 } ( f ( v ) ) = q _ { p + 1 } ( v )$.