Let $V$ be a vector subspace of $\mathscr{L}(E)$ included in $\operatorname{Sim}(E)$, of dimension $d \geqslant 1$. Show that there exists a vector subspace $W$ of $\mathscr{L}(E)$ included in $\operatorname{Sim}(E)$, of the same dimension $d$, and containing $\operatorname{Id}_{E}$.

Let $V$ be a vector subspace of $\mathscr{L}(E)$ containing $\operatorname{Id}_{E}$, included in $\operatorname{Sim}(E)$ and of dimension $d \geqslant 2$. Let $\left(\operatorname{Id}_{E}, f_{1}, \ldots, f_{d-1}\right)$ be a basis of $V$. Show that for all $i \in \{1, 2, \ldots, d-1\}$, $f_{i}^{*} + f_{i}$ is collinear to $\operatorname{Id}_{E}$.

Let $V$ be a vector subspace of $\mathscr{L}(E)$ containing $\operatorname{Id}_{E}$, included in $\operatorname{Sim}(E)$ and of dimension $d \geqslant 2$. Let $\left(\operatorname{Id}_{E}, f_{1}, \ldots, f_{d-1}\right)$ be a basis of $V$. Show that there exists a basis $\left(\operatorname{Id}_{E}, g_{1}, \ldots, g_{d-1}\right)$ of $V$ such that for all $i \in \{1, 2, \ldots, d-1\}$, $g_{i}$ is antisymmetric (one will seek $g_{i}$ as a combination of $f_{i}$ and $\operatorname{Id}_{E}$).

Let $V$ be a vector subspace of $\mathscr{L}(E)$ containing $\operatorname{Id}_{E}$, included in $\operatorname{Sim}(E)$ and of dimension $d \geqslant 2$. We fix a basis $\left(\operatorname{Id}_{E}, g_{1}, \ldots, g_{d-1}\right)$ of $V$ with for all $i$, $g_{i}$ antisymmetric. a) Show that for all $i \neq j$, $g_{i}g_{j} + g_{j}g_{i}$ is collinear to $\operatorname{Id}_{E}$. b) Show that we define a scalar product on $\mathscr{L}(E)$ by setting, for all $f, g$ of $\mathscr{L}(E)$: $(f \mid g) = \operatorname{tr}(f^{*}g)$. We consider, in the rest of this question, a basis $(h_{1}, \ldots, h_{d-1})$ of $\operatorname{Vect}(g_{1}, \ldots, g_{d-1})$ orthogonal for this scalar product. c) Show that the $h_{i}$ are antisymmetric and satisfy: $\forall i \neq j, h_{i}h_{j} + h_{j}h_{i} = 0$. What should be done so that the $h_{i}$ are also orthogonal automorphisms?

Conversely, let $(h_{1}, \ldots, h_{d-1})$ be a family of $\mathscr{L}(E)$ such that the $h_{i}$ are antisymmetric orthogonal automorphisms satisfying for all $i \neq j$, $h_{i}h_{j} + h_{j}h_{i} = 0$. Show that $\operatorname{Vect}\left(\operatorname{Id}_{E}, h_{1}, \ldots, h_{d-1}\right)$ is a vector subspace of $\mathscr{L}(E)$, of dimension $d$, included in $\operatorname{Sim}(E)$.

Let $p$ be an odd integer such that $\operatorname{dim}(E) = n = 2p$. We assume that there exist $d \geqslant 3$ and a family $(f_{1}, f_{2}, \ldots, f_{d-1})$ of elements of $\mathscr{L}(E)$ such that the $f_{i}$ are orthogonal automorphisms, antisymmetric satisfying: $\forall i \neq j, f_{i}f_{j} + f_{j}f_{i} = 0$. Let $x \in E$ of norm 1. a) Show that $(x, f_{1}(x), f_{2}(x), f_{1}f_{2}(x))$ is an orthonormal family, and that $S = \operatorname{Vect}(x, f_{1}(x), f_{2}(x), f_{1}f_{2}(x))$ is stable under $f_{1}$ and $f_{2}$. b) Deduce that $d_{n-4} \geqslant 3$.

In this question, $n = 2p$, with $p$ an odd integer. Show that $d_{n} = 2$.

In this section, the dimension of $E$ is 4. We assume that there exists a vector subspace of $\mathscr{L}(E)$ of dimension 4 included in $\operatorname{Sim}(E)$. We then consider, in accordance with I.D.4, a family $(f_{1}, f_{2}, f_{3})$ of elements of $\mathscr{L}(E)$ such that the $f_{i}$ are orthogonal automorphisms, antisymmetric satisfying: $\forall i \neq j, f_{i}f_{j} + f_{j}f_{i} = 0$. Let a fixed vector $x \in E$ of norm 1. a) Justify that the family $B = (x, f_{1}(x), f_{2}(x), f_{1}f_{2}(x))$ is a basis of $E$ then show that there exist real numbers $\alpha, \beta, \gamma, \delta$ such that: $$f_{3}(x) = \alpha x + \beta f_{1}(x) + \gamma f_{2}(x) + \delta f_{1}f_{2}(x)$$ Show that $\alpha = \beta = \gamma = 0$ and that $\delta \in \{-1, 1\}$. b) Show that $f_{3} = \delta f_{1}f_{2}$. If necessary, by replacing $f_{3}$ with its opposite, we assume in what follows that $f_{3} = f_{1}f_{2}$. c) If $x_{0}, x_{1}, x_{2}, x_{3}$ are real numbers, give the matrix $M(x_{0}, x_{1}, x_{2}, x_{3})$ in $B$ of the endomorphism $x_{0}\operatorname{Id}_{E} + x_{1}f_{1} + x_{2}f_{2} + x_{3}f_{3}$.

In this section, the dimension of $E$ is 4. Verify that for all $(x_{0}, x_{1}, x_{2}, x_{3}) \in \mathbb{R}^{4}$, $M(x_{0}, x_{1}, x_{2}, x_{3})$ is a similarity matrix. What can we conclude?

Show that $\mathcal { M } _ { 0 } ( n , \mathbb { K } )$ is a $\mathbb { K }$-vector space; specify its dimension.

Justify that, for every pair $(A , B)$ of elements of $\mathcal { M } ( n , \mathbb { K } )$, the matrix $[ A , B ]$ belongs to $\mathcal { M } _ { 0 } ( n , \mathbb { K } )$.

Show that the application $$\begin{aligned} j : & \mathbb { K } ^ { 3 } \longrightarrow \mathcal { M } _ { 0 } ( 2 , \mathbb { K } ) \\ \left( \begin{array} { l } x \\ y \\ z \end{array} \right) & \longmapsto \left( \begin{array} { c c } x & y + z \\ y - z & - x \end{array} \right) \end{aligned}$$ is an isomorphism of $\mathbb { K }$-vector spaces.

Let $A$ be a non-zero matrix of $\mathcal { M } _ { 0 } ( 2 , \mathbb { K } )$. Show that the following properties are equivalent:
i. The matrix $A$ is nilpotent;
ii. The spectrum of $A$ is equal to $\{ 0 \}$;
iii. The matrix $A$ is similar to the matrix $\left( \begin{array} { l l } 0 & 1 \\ 0 & 0 \end{array} \right)$.

We assume in this question that $\mathbb { K }$ is equal to $\mathbb { C }$.
Show that two non-zero matrices of $\mathcal { M } _ { 0 } ( 2 , \mathbb { C } )$ are similar if and only if they have the same characteristic polynomial.

We assume in this question that $\mathbb { K }$ is equal to $\mathbb { C }$.
Does the result that two non-zero matrices of $\mathcal { M } _ { 0 } ( 2 , \mathbb { C } )$ are similar if and only if they have the same characteristic polynomial remain true for two non-zero matrices of $\mathcal { M } _ { 0 } ( n , \mathbb { C } )$, with $n \geq 3$?

We assume in this question that $\mathbb { K }$ is equal to $\mathbb { R }$.
Let $A$ be a matrix of $\mathcal { M } _ { 0 } ( 2 , \mathbb { R } )$. We assume that its characteristic polynomial equals $X ^ { 2 } + r ^ { 2 }$, where $r$ is a non-zero real number.
a) Justify the existence of a matrix $P \in GL ( 2 , \mathbb { C } )$ satisfying: $ir H _ { 0 } = P ^ { - 1 } A P$. What is the value of the matrix $A ^ { 2 } + r ^ { 2 } I _ { 2 }$?
b) Let $f$ be the endomorphism of $\mathbb { R } ^ { 2 }$ canonically associated with the matrix $A$, that is, which maps a column vector $u$ of $\mathbb { R } ^ { 2 }$ to the vector $A u$. Let $w$ be a non-zero vector of $\mathbb { R } ^ { 2 }$. Prove that the family $\left( \frac { 1 } { r } f ( w ) , w \right)$ is a basis of $\mathbb { R } ^ { 2 }$, and give the matrix of $f$ in this basis.

We assume in this question that $\mathbb { K }$ is equal to $\mathbb { R }$.
Show that two non-zero matrices of $\mathcal { M } _ { 0 } ( 2 , \mathbb { R } )$ are similar in $\mathcal { M } ( 2 , \mathbb { R } )$ if and only if they have the same characteristic polynomial.

We assume in this question that $\mathbb { K }$ is equal to $\mathbb { R }$. We equip the vector space $\mathbb { R } ^ { 3 }$ with its canonical Euclidean affine structure and its canonical frame. For every matrix $A$ of $\mathcal { M } _ { 0 } ( 2 , \mathbb { R } )$, we denote by $\mathcal { Q } _ { A }$ the set of points of $\mathbb { R } ^ { 3 }$ whose image by the application $j$ has the same characteristic polynomial as $A$.
a) Let $r$ be a strictly positive real number. Show that each of the sets $\mathcal { Q } _ { X _ { 0 } } , \mathcal { Q } _ { r J _ { 0 } }$ and $\mathcal { Q } _ { r H _ { 0 } }$ is a quadric for which an equation will be specified.
b) Draw graphically the appearance of the quadrics $\mathcal { Q } _ { X _ { 0 } } , \mathcal { Q } _ { J _ { 0 } }$ and $\mathcal { Q } _ { H _ { 0 } }$ on the same drawing.

Let $A , B$ and $M$ be three elements of $\mathcal { M } _ { 0 } ( 2 , \mathbb { K } )$.
Express the trace of the matrix $M ^ { 2 }$ in terms of the determinant of $M$.

Let $A , B$ and $M$ be three elements of $\mathcal { M } _ { 0 } ( 2 , \mathbb { K } )$.
Prove that the matrix $M$ is nilpotent if and only if the trace of the matrix $M ^ { 2 }$ is zero.

Let $A , B$ and $M$ be three elements of $\mathcal { M } _ { 0 } ( 2 , \mathbb { K } )$. We assume that the matrices $A$ and $[ A , B ]$ commute.
Prove that the matrix $[ A , B ]$ is nilpotent.

Determine the matrices $M$ of $\mathcal { M } ( 2 , \mathbb { K } )$ that commute with $X _ { 0 }$. What are the matrices $M$ of $\mathcal { M } _ { 0 } ( 2 , \mathbb { K } )$ that commute with $X _ { 0 }$?
Recall that $X _ { 0 } = \left( \begin{array} { l l } 0 & 1 \\ 0 & 0 \end{array} \right)$, $H _ { 0 } = \left( \begin{array} { c c } 1 & 0 \\ 0 & - 1 \end{array} \right)$, $Y _ { 0 } = \left( \begin{array} { l l } 0 & 0 \\ 1 & 0 \end{array} \right)$, $J _ { 0 } = \left( \begin{array} { c c } 0 & 1 \\ - 1 & 0 \end{array} \right)$.

Let $P$ be a matrix of $GL ( 2 , \mathbb { K } )$. Verify that $( P X _ { 0 } P ^ { - 1 } , P H _ { 0 } P ^ { - 1 } , P Y _ { 0 } P ^ { - 1 } )$ is an admissible triple.
Recall that a triple $(X, H, Y)$ of three non-zero matrices of $\mathcal{M}(n, \mathbb{K})$ is an admissible triple if $[H,X]=2X$, $[X,Y]=H$, $[H,Y]=-2Y$.

Let $X, H, Y$ be three elements of $\mathcal { M } _ { 0 } ( 2 , \mathbb { K } )$ such that $(X, H, Y)$ forms an admissible triple (i.e., $[H,X]=2X$, $[X,Y]=H$, $[H,Y]=-2Y$).
Show using questions II.F and II.C that there exists a matrix $Q \in GL ( 2 , \mathbb { K } )$ satisfying $X = Q X _ { 0 } Q ^ { - 1 }$.

Let $X, H, Y$ be three elements of $\mathcal { M } _ { 0 } ( 2 , \mathbb { K } )$ such that $(X, H, Y)$ forms an admissible triple. Fix a matrix $Q \in GL(2, \mathbb{K})$ satisfying $X = QX_0Q^{-1}$. We define the vectors $u = Q \binom{1}{0}$ and $v = Q \binom{0}{1}$.
a) By computing the vector $[ H , X ] u$ in two different ways, prove that $u$ is an eigenvector of the matrix $H$.
b) By computing the vector $[ H , X ] v$ in two different ways, prove the existence of a scalar $t$ satisfying the identity: $H = Q \left( \begin{array} { c c } 1 & t \\ 0 & - 1 \end{array} \right) Q ^ { - 1 }$.
c) Find a matrix $T \in GL ( 2 , \mathbb { K } )$ commuting with $X _ { 0 }$ and satisfying the relation $H = Q T H _ { 0 } ( Q T ) ^ { - 1 }$.