Let $E$ be a $\mathbb{C}$-vector space of dimension $n \geq 1$. We say that an endomorphism $u$ of $E$ is a permutation endomorphism if there exists a basis $(e_1, \ldots, e_n)$ of $E$ and a permutation $\sigma \in \mathfrak{S}_n$ such that $u(e_j) = e_{\sigma(j)}$ for all $j \in \llbracket 1, n \rrbracket$.
Show that $u$ is a permutation endomorphism if and only if there exists a basis in which its matrix is a permutation matrix.

Let $E$ be a $\mathbb{C}$-vector space of dimension $n \geq 1$. Let $u$ be an endomorphism of $E$ such that $u^2 = \operatorname{Id}_E$. Show that $u$ is a permutation endomorphism if and only if $\operatorname{Tr}(u)$ is a natural integer.

Let $E$ be a $\mathbb{C}$-vector space of dimension $n \geq 1$. Study whether the equivalence of the previous question holds when we replace the hypothesis $u^2 = \operatorname{Id}_E$ by $u^k = \operatorname{Id}_E$ for $k = 3$, then for $k = 4$.

Let $E$ be a $\mathbb{C}$-vector space of dimension $n \geq 1$. Let $u$ be an endomorphism of $E$. Show that $u$ is a permutation endomorphism if and only if it satisfies the following two conditions:
(a) there exist natural integers $c_1, \ldots, c_n$ such that $\chi_u = \prod_{\ell=1}^{n} \left(X^\ell - 1\right)^{c_\ell}$;
(b) there exists $N$ such that $u^N = \operatorname{Id}_E$.

Let $M \in \mathcal{A}_{2n}(\mathbb{R}) \cap \mathrm{Sp}_{2n}(\mathbb{R})$, and let $m$ be the linear map canonically associated with $M$. Let $X$ denote an eigenvector of $M^{2}$ of norm 1 associated with a certain eigenvalue $\lambda$. Denote $F = \operatorname{Vect}(X, MX, J_{n} X, J_{n} MX)$. Show that $F$ is stable under $M$ and under $J_{n}$.

Let $M \in \mathcal{A}_{2n}(\mathbb{R}) \cap \mathrm{Sp}_{2n}(\mathbb{R})$, and let $m$ be the linear map canonically associated with $M$. Show that there exists a non-zero natural integer $q$ and vector subspaces of $\mathcal{M}_{2n,1}(\mathbb{R})$, denoted $F_{1}, \ldots, F_{q}$ such that

• [(a)] $F_{1} \oplus \cdots \oplus F_{q} = \mathcal{M}_{2n,1}(\mathbb{R})$;
• [(b)] $\forall i \in \{1,\ldots,q\}$, $F_{i}$ is stable under $M$ and under $J_{n}$;
• [(c)] $\forall i \in \{1,\ldots,q\}$, $F_{i}^{\perp}$ is stable under $M$ and under $J_{n}$;
• [(d)] $\forall (i,j) \in \{1,\ldots,q\}^{2}$, $i \neq j \Longrightarrow \forall (Y,Z) \in F_{i} \times F_{j}$, $\langle Y,Z \rangle = 0 = \varphi(Y,Z)$;
• [(e)] $\forall i \in \{1,\ldots,q\}$, $\dim F_{i} \in \{2,4\}$;
• [(f)] $\forall i \in \{1,\ldots,q\}$, the matrix of the application $m_{F_{i}}$ induced by $m$ on $F_{i}$ in a certain basis is of the form $$J_{1} \quad \text{or} \quad \left(\begin{array}{cc} \sqrt{-\lambda} J_{1} & 0_{2,2} \\ 0_{2,2} & \frac{1}{\sqrt{-\lambda}} J_{1} \end{array}\right).$$

Let $F$ be a vector subspace of $E$ stable under $u$. Show that the orthogonal complement $F^{\perp}$ of $F$ is stable under $u$.

Let $(E, \langle \cdot, \cdot \rangle)$ be a real pre-Hilbert space, with associated norm $\|\cdot\|$. Let $u$ be an endomorphism of $E$ satisfying, $$\forall (x,y) \in E^2, \quad \langle u(x), y \rangle = \langle x, u(y) \rangle$$ Suppose that there exists a unit vector $x_0 \in F$ satisfying $$\langle u(x_0), x_0 \rangle = \sup_{x \in F, \|x\|=1} \langle u(x), x \rangle$$ For every unit vector $y \in F$ orthogonal to $x_0$, we set, for every real $t$, $$\begin{aligned} & \gamma(t) = x_0 \cos t + y \sin t \\ & \varphi(t) = \langle u \circ \gamma(t), \gamma(t) \rangle \end{aligned}$$ Show that $\varphi$ is of class $\mathcal{C}^1$.

Let $(E, \langle \cdot, \cdot \rangle)$ be a real pre-Hilbert space, with associated norm $\|\cdot\|$. Let $u$ be an endomorphism of $E$ satisfying, $$\forall (x,y) \in E^2, \quad \langle u(x), y \rangle = \langle x, u(y) \rangle$$ Suppose that there exists a unit vector $x_0 \in F$ satisfying $$\langle u(x_0), x_0 \rangle = \sup_{x \in F, \|x\|=1} \langle u(x), x \rangle$$ For every unit vector $y \in F$ orthogonal to $x_0$, we set, for every real $t$, $$\begin{aligned} & \gamma(t) = x_0 \cos t + y \sin t \\ & \varphi(t) = \langle u \circ \gamma(t), \gamma(t) \rangle \end{aligned}$$ Calculate $\|\gamma(t)\|$ then justify that $\varphi'(0) = 0$.

Let $(E, \langle \cdot, \cdot \rangle)$ be a real pre-Hilbert space, with associated norm $\|\cdot\|$. Let $u$ be an endomorphism of $E$ satisfying, $$\forall (x,y) \in E^2, \quad \langle u(x), y \rangle = \langle x, u(y) \rangle$$ Suppose that there exists a unit vector $x_0 \in F$ satisfying $$\langle u(x_0), x_0 \rangle = \sup_{x \in F, \|x\|=1} \langle u(x), x \rangle$$ For every unit vector $y \in F$ orthogonal to $x_0$, we set, for every real $t$, $$\begin{aligned} & \gamma(t) = x_0 \cos t + y \sin t \\ & \varphi(t) = \langle u \circ \gamma(t), \gamma(t) \rangle \end{aligned}$$ Deduce that $u(x_0)$ is orthogonal to $y$.

Let $u \in \mathcal{N}(E)$. Show that $\operatorname{tr} u^{k} = 0$ for every $k \in \mathbf{N}^{*}$.

We fix a basis $\mathbf{B}$ of $E$. We denote by $\mathcal{N}_{\mathbf{B}}$ the set of endomorphisms of $E$ whose matrix in $\mathbf{B}$ is strictly upper triangular. Justify that $\mathcal{N}_{\mathbf{B}}$ is a nilpotent vector subspace of $\mathcal{L}(E)$ and that its dimension equals $\frac{n(n-1)}{2}$.

Let $\mathbf{B}$ be a basis of $E$. Show that
$$\left\{\nu(u) \mid u \in \mathcal{N}_{\mathbf{B}}\right\} = \{\nu(u) \mid u \in \mathcal{N}(E)\} = \llbracket 1, n \rrbracket$$

Let $u \in \mathcal{L}(E)$. We are given two vectors $x$ and $y$ of $E$, as well as two integers $p \geq q \geq 1$ such that $u^{p}(x) = u^{q}(y) = 0$ and $u^{p-1}(x) \neq 0$. Show that the family $(x, u(x), \ldots, u^{p-1}(x))$ is free, and that if $(u^{p-1}(x), u^{q-1}(y))$ is free then $(x, u(x), \ldots, u^{p-1}(x), y, u(y), \ldots, u^{q-1}(y))$ is free.

Let $u \in \mathcal{N}(E)$, with nilindex $p$. Deduce from the previous question that if $p \geq n-1$ and $p \geq 2$ then $\operatorname{Im} u^{p-1} = \operatorname{Im} u \cap \operatorname{Ker} u$ and $\operatorname{Im} u^{p-1}$ has dimension 1.

We consider a Euclidean vector space $(E, (-\mid-))$. Given $a \in E$ and $x \in E$, we denote by $a \otimes x$ the map from $E$ to itself defined by:
$$\forall z \in E, (a \otimes x)(z) = (a \mid z) \cdot x$$
We fix $x \in E \backslash \{0\}$. Show that the map $a \in E \mapsto a \otimes x$ is linear and constitutes a bijection from $E$ onto $\{u \in \mathcal{L}(E) : \operatorname{Im} u \subset \operatorname{Vect}(x)\}$.

We consider a Euclidean vector space $(E, (-\mid-))$. Given $a \in E$ and $x \in E$, we denote by $a \otimes x$ the map from $E$ to itself defined by:
$$\forall z \in E, (a \otimes x)(z) = (a \mid z) \cdot x$$
Let $a \in E$ and $x \in E \backslash \{0\}$. Show that $\operatorname{tr}(a \otimes x) = (a \mid x)$.

We consider an $\mathbf{R}$-vector space $E$ of dimension $n > 0$. Let $\mathcal{V}$ be a nilpotent vector subspace of $\mathcal{L}(E)$ containing a non-zero element, with generic nilindex $p := \max_{u \in \mathcal{V}} \nu(u)$. We introduce the subset $\mathcal{V}^{\bullet}$ of $E$ formed by vectors belonging to at least one of the sets $\operatorname{Im} u^{p-1}$ for $u$ in $\mathcal{V}$, the vector subspace $K(\mathcal{V}) := \operatorname{Vect}(\mathcal{V}^{\bullet})$, and given $x \in E$, $\mathcal{V} x := \{v(x) \mid v \in \mathcal{V}\}$.
Lemma B states: Let $x$ be in $\mathcal{V}^{\bullet} \backslash \{0\}$. If $K(\mathcal{V}) \subset \operatorname{Vect}(x) + \mathcal{V} x$, then $v(x) = 0$ for every $v$ in $\mathcal{V}$.
Let $x \in \mathcal{V}^{\bullet} \backslash \{0\}$ such that $K(\mathcal{V}) \subset \operatorname{Vect}(x) + \mathcal{V} x$. We choose $u \in \mathcal{V}$ such that $x \in \operatorname{Im} u^{p-1}$.
Given $y \in K(\mathcal{V})$, show that for every $k \in \mathbf{N}$ there exist $y_{k} \in K(\mathcal{V})$ and $\lambda_{k} \in \mathbf{R}$ such that $y = \lambda_{k} x + u^{k}(y_{k})$. Deduce that $K(\mathcal{V}) \subset \operatorname{Vect}(x)$ and then that $v(x) = 0$ for every $v \in \mathcal{V}$.

We fix a real vector space $E$ of dimension $n$, as well as a nilpotent vector subspace $\mathcal{V}$ of $\mathcal{L}(E)$, equipped with an inner product $(-\mid-)$. We consider an arbitrary vector $x$ of $E \backslash \{0\}$, and set
$$H := \operatorname{Vect}(x)^{\perp}, \quad \mathcal{V} x := \{v(x) \mid v \in \mathcal{V}\} \text{ and } \mathcal{W} := \{v \in \mathcal{V} : v(x) = 0\}$$
We denote by $\pi$ the orthogonal projection of $E$ onto $H$. For $u \in \mathcal{W}$, we denote by $\bar{u}$ the endomorphism of $H$ defined by $\forall z \in H, \bar{u}(z) = \pi(u(z))$. We consider the sets $\overline{\mathcal{V}} := \{\bar{u} \mid u \in \mathcal{W}\}$ and $\mathcal{Z} := \{u \in \mathcal{W} : \bar{u} = 0\}$.
Show that $\mathcal{V} x$, $\mathcal{W}$, $\overline{\mathcal{V}}$ and $\mathcal{Z}$ are vector subspaces of $E$, $\mathcal{V}$, $\mathcal{L}(H)$ and $\mathcal{V}$ respectively.

We fix a real vector space $E$ of dimension $n$, as well as a nilpotent vector subspace $\mathcal{V}$ of $\mathcal{L}(E)$, equipped with an inner product $(-\mid-)$. We consider an arbitrary vector $x$ of $E \backslash \{0\}$, and set
$$H := \operatorname{Vect}(x)^{\perp}, \quad \mathcal{V} x := \{v(x) \mid v \in \mathcal{V}\} \text{ and } \mathcal{W} := \{v \in \mathcal{V} : v(x) = 0\}$$
We denote by $\pi$ the orthogonal projection of $E$ onto $H$. For $u \in \mathcal{W}$, we denote by $\bar{u}$ the endomorphism of $H$ defined by $\forall z \in H, \bar{u}(z) = \pi(u(z))$. We consider the sets $\overline{\mathcal{V}} := \{\bar{u} \mid u \in \mathcal{W}\}$ and $\mathcal{Z} := \{u \in \mathcal{W} : \bar{u} = 0\}$. Given $a \in E$ and $x \in E$, $(a \otimes x)(z) = (a \mid z) \cdot x$ for all $z \in E$.
Show that there exists a vector subspace $L$ of $E$ such that
$$\mathcal{Z} = \{a \otimes x \mid a \in L\} \quad \text{and} \quad \operatorname{dim} L = \operatorname{dim} \mathcal{Z},$$
and show that then $x \in L^{\perp}$.

We fix a real vector space $E$ of dimension $n$, as well as a nilpotent vector subspace $\mathcal{V}$ of $\mathcal{L}(E)$, equipped with an inner product $(-\mid-)$. We consider an arbitrary vector $x$ of $E \backslash \{0\}$, and set
$$H := \operatorname{Vect}(x)^{\perp}, \quad \mathcal{V} x := \{v(x) \mid v \in \mathcal{V}\} \text{ and } \mathcal{W} := \{v \in \mathcal{V} : v(x) = 0\}$$
We consider the sets $\overline{\mathcal{V}} := \{\bar{u} \mid u \in \mathcal{W}\}$ and $\mathcal{Z} := \{u \in \mathcal{W} : \bar{u} = 0\}$. Given $a \in E$ and $x \in E$, $(a \otimes x)(z) = (a \mid z) \cdot x$ for all $z \in E$. There exists a vector subspace $L$ of $E$ such that $\mathcal{Z} = \{a \otimes x \mid a \in L\}$ and $x \in L^{\perp}$.
By considering $u$ and $a \otimes x$ for $u \in \mathcal{V}$ and $a \in L$, deduce from Lemma A that $\mathcal{V} x \subset L^{\perp}$, and that more generally $u^{k}(x) \in L^{\perp}$ for every $k \in \mathbf{N}$ and every $u \in \mathcal{V}$.

We fix a real vector space $E$ of dimension $n$, as well as a nilpotent vector subspace $\mathcal{V}$ of $\mathcal{L}(E)$, equipped with an inner product $(-\mid-)$. We consider an arbitrary vector $x$ of $E \backslash \{0\}$, and set
$$H := \operatorname{Vect}(x)^{\perp}, \quad \mathcal{V} x := \{v(x) \mid v \in \mathcal{V}\}$$
There exists a vector subspace $L$ of $E$ such that $\mathcal{Z} = \{a \otimes x \mid a \in L\}$, $x \in L^{\perp}$, and $\mathcal{V} x \subset L^{\perp}$.
Justify that $\lambda x \notin \mathcal{V} x$ for every $\lambda \in \mathbf{R}^{*}$, and deduce from the two previous questions that
$$\operatorname{dim} \mathcal{V} x + \operatorname{dim} L \leq n-1$$

We fix a real vector space $E$ of dimension $n$, as well as a nilpotent vector subspace $\mathcal{V}$ of $\mathcal{L}(E)$, equipped with an inner product $(-\mid-)$. We consider an arbitrary vector $x$ of $E \backslash \{0\}$, and set
$$H := \operatorname{Vect}(x)^{\perp}, \quad \mathcal{W} := \{v \in \mathcal{V} : v(x) = 0\}$$
We denote by $\pi$ the orthogonal projection of $E$ onto $H$. For $u \in \mathcal{W}$, we denote by $\bar{u}$ the endomorphism of $H$ defined by $\forall z \in H, \bar{u}(z) = \pi(u(z))$. We consider the set $\overline{\mathcal{V}} := \{\bar{u} \mid u \in \mathcal{W}\}$.
Let $u \in \mathcal{W}$. Show that $(\bar{u})^{k}(z) = \pi(u^{k}(z))$ for every $k \in \mathbf{N}$ and every $z \in H$. Deduce that $\overline{\mathcal{V}}$ is a nilpotent vector subspace of $\mathcal{L}(H)$.

We fix a real vector space $E$ of dimension $n \geq 2$, as well as a nilpotent vector subspace $\mathcal{V}$ of $\mathcal{L}(E)$, equipped with an inner product $(-\mid-)$. We consider an arbitrary vector $x$ of $E \backslash \{0\}$, and set $H := \operatorname{Vect}(x)^{\perp}$, $\mathcal{V} x := \{v(x) \mid v \in \mathcal{V}\}$, $\mathcal{W} := \{v \in \mathcal{V} : v(x) = 0\}$, $\overline{\mathcal{V}} := \{\bar{u} \mid u \in \mathcal{W}\}$, $\mathcal{Z} := \{u \in \mathcal{W} : \bar{u} = 0\}$, and $L$ the vector subspace such that $\mathcal{Z} = \{a \otimes x \mid a \in L\}$.
We have $\operatorname{dim} \mathcal{V} = \operatorname{dim}(\mathcal{V} x) + \operatorname{dim} \mathcal{Z} + \operatorname{dim} \overline{\mathcal{V}}$, $\operatorname{dim} \mathcal{V} x + \operatorname{dim} L \leq n-1$, $\overline{\mathcal{V}}$ is a nilpotent subspace of $\mathcal{L}(H)$ with $\dim H = n-1$, and by induction hypothesis $\operatorname{dim} \overline{\mathcal{V}} \leq \frac{(n-1)(n-2)}{2}$.
Prove that
$$\operatorname{dim} \mathcal{V} \leq \frac{n(n-1)}{2}$$

Suppose that the sequence of vectors $\left( P ^ { ( k ) } \right) _ { k \in \mathbb { N } }$ converges to a vector $P = \left( p _ { 1 } , \ldots , p _ { n } \right)$. Show that $P T = P$, that for all $i \in \llbracket 1 , n \rrbracket$, $p _ { i } \geqslant 0$ and that $p _ { 1 } + \cdots + p _ { n } = 1$.