We fix a $\mathbb{C}$-vector space $E$ of dimension $n \geqslant 2$. Let $\mathcal{A}$ be an irreducible subalgebra of $\mathcal{L}(E)$ (i.e., the only vector subspaces stable by all elements of $\mathcal{A}$ are $\{0\}$ and $E$). Let $u_{0} \in \mathcal{A}$ be of rank 1. We can therefore choose a basis $\mathcal{B} = (\varepsilon_{1}, \ldots, \varepsilon_{n})$ of $E$ such that $(\varepsilon_{2}, \ldots, \varepsilon_{n})$ is a basis of $\ker u_{0}$.
Show that there exist $u_{1}, \ldots, u_{n} \in \mathcal{A}$ of rank 1 such that $u_{i}(\varepsilon_{1}) = \varepsilon_{i}$ for all $i \in \llbracket 1, n \rrbracket$.

We fix a $\mathbb{C}$-vector space $E$ of dimension $n \geqslant 2$. Let $\mathcal{A}$ be an irreducible subalgebra of $\mathcal{L}(E)$ (i.e., the only vector subspaces stable by all elements of $\mathcal{A}$ are $\{0\}$ and $E$). Let $u_{0} \in \mathcal{A}$ be of rank 1. We can therefore choose a basis $\mathcal{B} = (\varepsilon_{1}, \ldots, \varepsilon_{n})$ of $E$ such that $(\varepsilon_{2}, \ldots, \varepsilon_{n})$ is a basis of $\ker u_{0}$.
Conclude (that $\mathcal{A} = \mathcal{L}(E)$).

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 $$J_{1} \quad \text{or} \quad \left(\begin{array}{cc} \sqrt{-\lambda} J_{1} & 0_{2,2} \\ 0_{2,2} & \frac{1}{\sqrt{-\lambda}} J_{1} \end{array}\right).$$

In this subsection, $E$ is 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.

In this subsection, $E$ is 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 $j \in \llbracket 1, n \rrbracket$.
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 $u^N = \operatorname{Id}_E$.

In this subsection, $E$ is 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 $j \in \llbracket 1, n \rrbracket$.
Let $u$ be a diagonalizable endomorphism of $E$. Show that $u$ is a permutation endomorphism if and only if there exist natural integers $c_1, \ldots, c_n$ such that, for all $k \in \mathbb{N}$,
$$\operatorname{Tr}\left(u^k\right) = \sum_{\substack{\ell=1 \\ \ell \mid k}}^{n} \ell c_\ell$$
(We sum over the values of $\ell$ dividing $k$ and belonging to $\llbracket 1, n \rrbracket$.)

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 $F$ 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 all 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 all 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 all 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 $\{u \in \mathcal{L}(E) : \operatorname{Im} u \subset \operatorname{Vect}(x)\}$.