Let $V$ be a $\mathbb { K }$-vector space of finite non-zero dimension. Let $f$ and $g$ be two endomorphisms of $V$ that commute, that is, such that $f \circ g = g \circ f$. Show that the eigenspaces of $f$ are stable under $g$.

Let $\mathcal { A }$ be a non-zero vector subspace of $\mathcal { M } ( n , \mathbb { K } )$ stable by bracket, and let $\mathcal { E }$ be the intersection of $\mathcal { A }$ and $\mathcal { D } ( n , \mathbb { K } )$. For every map $\lambda$ from $\mathcal { E }$ to $\mathbb { K }$, we set: $$\mathcal { A } _ { \lambda } = \left\{ M \in \mathcal { A } \mid \Phi _ { H } ( M ) = \lambda ( H ) M \text { for all } H \in \mathcal { E } \right\}$$
Let $\lambda$ be a map from $\mathcal { E }$ to $\mathbb { K }$.
a) Show that $\mathcal { A } _ { \lambda }$ is a vector subspace of $\mathcal { A }$.
b) Show that if $\mathcal { A } _ { \lambda }$ is not reduced to $\{ 0 \}$, then $\lambda$ is a linear form on $\mathcal { E }$.

We set $\xi_i^j = \begin{cases}0 & \text{if } i = j \\ 1 & \text{otherwise}\end{cases}$ and $N_k = (m_{ij} + k\xi_i^j)$ with $k$ a strictly positive real number.
Let $A \in \mathcal{M}_n(\mathbb{R})$. Show that the hyperplane $\mathcal{H}$ with normal vector $Z$ (and equation $x_1 + \cdots + x_n = 0$) is stable under the canonical endomorphism associated with the matrix $\Psi(A)$.

Let $s$ and $t$ be two symmetries of $E$ that anticommute, that is, such that $s \circ t + t \circ s = 0$. a) Prove the equalities $t(F_s) = G_s$ and $t(G_s) = F_s$. b) Deduce that $F_s$ and $G_s$ have the same dimension and that $n$ is even.

In this problem, $\mathbb{K}$ denotes the field $\mathbb{R}$ or the field $\mathbb{C}$ and $E$ is a non-zero $\mathbb{K}$-vector space. If $f$ is an endomorphism of $E$, for every subspace $F$ of $E$ stable by $f$ we denote by $f_F$ the endomorphism of $F$ induced by $f$.
Show that a line $F$ generated by a vector $u$ is stable by $f$ if and only if $u$ is an eigenvector of $f$.

In this problem, $\mathbb{K}$ denotes the field $\mathbb{R}$ or the field $\mathbb{C}$ and $E$ is a non-zero $\mathbb{K}$-vector space. $f$ is an endomorphism of a $\mathbb{K}$-vector space $E$.
I.B.1) Show that there exist at least two subspaces of $E$ stable by $f$ and give an example of an endomorphism of $\mathbb{R}^2$ which admits only two stable subspaces.
I.B.2) Show that if $E$ is of finite dimension $n \geqslant 2$ and if $f$ is non-zero and non-injective, then there exist at least three subspaces of $E$ stable by $f$ and at least four when $n$ is odd.
Give an example of an endomorphism of $\mathbb{R}^2$ which admits only three stable subspaces.

In this problem, $\mathbb{K}$ denotes the field $\mathbb{R}$ or the field $\mathbb{C}$ and $E$ is a non-zero $\mathbb{K}$-vector space. $f$ is an endomorphism of a $\mathbb{K}$-vector space $E$.
I.C.1) Show that every subspace generated by a family of eigenvectors of $f$ is stable by $f$. Specify the endomorphism induced by $f$ on every eigenspace of $f$.
I.C.2) Show that if $f$ admits an eigenspace of dimension at least equal to 2 then there exist infinitely many lines of $E$ stable by $f$.
I.C.3) What can be said about $f$ if all subspaces of $E$ are stable by $f$?

In this problem, $\mathbb{K}$ denotes the field $\mathbb{R}$ or the field $\mathbb{C}$ and $E$ is a non-zero $\mathbb{K}$-vector space. $f$ is an endomorphism of a $\mathbb{K}$-vector space $E$. In this subsection, $E$ is a space of finite dimension.
I.D.1) Show that if $f$ is diagonalisable then every subspace of $E$ admits a complement in $E$ stable by $f$.
One may start from a basis of $F$ and a basis of $E$ consisting of eigenvectors of $f$.
I.D.2) Show that if $\mathbb{K} = \mathbb{C}$ and if every subspace of $E$ stable by $f$ admits a complement in $E$ stable by $f$, then $f$ is diagonalisable.
What about the case if $\mathbb{K} = \mathbb{R}$?

In this part, $n$ and $p$ are two natural integers at least equal to 2, $f$ is a diagonalisable endomorphism of a $\mathbb{K}$-vector space $E$ of dimension $n$, which admits $p$ distinct eigenvalues $\{\lambda_1, \ldots, \lambda_p\}$ and, for all $i$ in $\llbracket 1, p \rrbracket$, we denote by $E_i$ the eigenspace of $f$ associated with the eigenvalue $\lambda_i$.
The goal here is to show that a subspace $F$ of $E$ is stable by $f$ if and only if $F = \bigoplus_{i=1}^{p} (F \cap E_i)$.
II.A.1) Show that every subspace $F$ of $E$ such that $F = \bigoplus_{i=1}^{p} (F \cap E_i)$ is stable by $f$.
II.A.2) Let $F$ be a subspace of $E$ stable by $f$ and $x$ a non-zero vector of $F$.
Justify the existence and uniqueness of $(x_i)_{1 \leqslant i \leqslant p}$ in $E_1 \times \cdots \times E_p$ such that $x = \sum_{i=1}^{p} x_i$.
II.A.3) If we denote $H_x = \{i \in \llbracket 1, p \rrbracket \mid x_i \neq 0\}$, $H_x$ is non-empty and, up to reordering the eigenvalues (and the eigenspaces), we can assume that $H_x = \llbracket 1, r \rrbracket$ with $1 \leqslant r \leqslant p$. Thus we have $x = \sum_{i=1}^{r} x_i$ with $x_i \in E_i \setminus \{0\}$ for all $i$ in $\llbracket 1, r \rrbracket$.
We denote $V_x = \operatorname{Vect}(x_1, \ldots, x_r)$.
Show that $\mathcal{B}_x = (x_1, \ldots, x_r)$ is a basis of $V_x$.
II.A.4) Show that for all $j$ in $\llbracket 1, r \rrbracket$, $f^{j-1}(x)$ belongs to $V_x$ and give the matrix of the family $(f^{j-1}(x))_{1 \leqslant j \leqslant r}$ in the basis $\mathcal{B}_x$.
II.A.5) Show that $(f^{j-1}(x))_{1 \leqslant j \leqslant r}$ is a basis of $V_x$.
II.A.6) Deduce that for all $i$ in $\llbracket 1, p \rrbracket$, $x_i$ belongs to $F$ and conclude.

In this part, $n$ and $p$ are two natural integers at least equal to 2, $f$ is a diagonalisable endomorphism of a $\mathbb{K}$-vector space $E$ of dimension $n$, which admits $p$ distinct eigenvalues $\{\lambda_1, \ldots, \lambda_p\}$ and, for all $i$ in $\llbracket 1, p \rrbracket$, we denote by $E_i$ the eigenspace of $f$ associated with the eigenvalue $\lambda_i$. In this subsection, we place ourselves in the case where $p = n$.
II.B.1) Specify the dimension of $E_i$ for all $i$ in $\llbracket 1, p \rrbracket$.
II.B.2) How many lines of $E$ are stable by $f$?
II.B.3) If $n \geqslant 3$ and $k \in \llbracket 2, n-1 \rrbracket$, how many subspaces of $E$ of dimension $k$ and stable by $f$ are there?
II.B.4) How many subspaces of $E$ are stable by $f$ in this case? Give them all.

We consider the differentiation endomorphism $D$ on $\mathbb{K}[X]$ defined by $D(P) = P'$ for all $P$ in $\mathbb{K}[X]$.
III.A.1) Verify that for all $n$ in $\mathbb{N}$, $\mathbb{K}_n[X]$ is stable by $D$ and give the matrix $A_n$ of the endomorphism induced by $D$ on $\mathbb{K}_n[X]$ in the canonical basis of $\mathbb{K}_n[X]$.
III.A.2) Let $F$ be a subspace of $\mathbb{K}[X]$, of finite non-zero dimension, stable by $D$.
a) Justify the existence of a natural integer $n$ and a polynomial $R$ of degree $n$ such that $R \in F$ and $F \subset \mathbb{K}_n[X]$.
b) Show that the family $(D^i(R))_{0 \leqslant i \leqslant n}$ is a free family of $F$.
c) Deduce that $F = \mathbb{K}_n[X]$.
III.A.3) Give all subspaces of $\mathbb{K}[X]$ stable by $D$.

We consider an endomorphism $f$ of a $\mathbb{K}$-vector space $E$ of dimension $n \geqslant 2$ such that $f^n = 0$ and $f^{n-1} \neq 0$.
III.B.1) Determine the set of vectors $u$ of $E$ such that the family $\mathcal{B}_{f,u} = (f^{n-i}(u))_{1 \leqslant i \leqslant n}$ is a basis of $E$.
III.B.2) In the case where $\mathcal{B}_{f,u}$ is a basis of $E$, what is the matrix of $f$ in $\mathcal{B}_{f,u}$?
III.B.3) Determine a basis of $E$ such that the matrix of $f$ in this basis is $A_{n-1}$.
III.B.4) Give all subspaces of $E$ stable by $f$. How many are there? Give a simple relation between these stable subspaces and the kernels $\ker(f^i)$ for $i$ in $\llbracket 0, n \rrbracket$.

In this part, $n$ is a non-zero natural integer, $M$ is in $\mathcal{M}_n(\mathbb{R})$ and $f$ is the endomorphism of $E = \mathcal{M}_{n,1}(\mathbb{R})$ defined by $f(X) = MX$ for all $X$ in $E$.
What do you think of the statement: ``every endomorphism of a finite-dimensional real vector space admits at least one line or one plane stable''?

Does there exist an endomorphism of $\mathbb{R}[X]$ admitting neither line nor plane stable?

In all that follows, $\mathbf{K = C}$. We set $E = \mathbf{C}^n$. We are given $A \in \mathcal{M}_n(\mathbf{C})$. We introduce the following polynomials: $$\begin{aligned} & P_s(X) = \prod_{\substack{\lambda \in \operatorname{Sp}(A) \\ \operatorname{Re}(\lambda) < 0}} (X - \lambda)^{m_\lambda}, \\ & P_i(X) = \prod_{\substack{\lambda \in \operatorname{Sp}(A) \\ \operatorname{Re}(\lambda) > 0}} (X - \lambda)^{m_\lambda}, \\ & P_n(X) = \prod_{\substack{\lambda \in \operatorname{Sp}(A) \\ \operatorname{Re}(\lambda) = 0}} (X - \lambda)^{m_\lambda}, \end{aligned}$$ and the subspaces $E_s = \operatorname{Ker}(P_s(A))$, $E_i = \operatorname{Ker}(P_i(A))$ and $E_n = \operatorname{Ker}(P_n(A))$ of $E = \mathbf{C}^n$.
$\mathbf{25}$ ▷ After justifying that $E = E_s \oplus E_i \oplus E_n$, show that $$E_s = \left\{ X \in E \mid \lim_{t \rightarrow +\infty} e^{tA} X = 0 \right\}.$$

We fix an integer $d \in \mathbb{N}^*$, and we equip $\mathbb{R}^d$ with the usual inner product denoted $\langle \cdot, \cdot \rangle$ and the associated Euclidean norm $\|\cdot\|$. We denote $C := \{x \in \mathbb{R}^d \mid \|x\| \leq 1\}$ the closed unit ball of $\mathbb{R}^d$. Let $M$ be a nonzero real symmetric matrix of size $d \times d$ such that $\forall x \in \mathbb{R}^d,\, \langle x, Mx \rangle \geq 0$. We define $f(x) := -\frac{1}{2}\langle x, Mx \rangle$, so that $\nabla f(x) = -Mx$. The sequence $(x_n)_{n \in \mathbb{N}}$ is defined by $$x_{n+1} := P_C(x_n - \tau \nabla f(x_n)), \quad \text{with} \quad P_C(x) := \begin{cases} x & \text{if } \|x\| \leq 1, \\ x/\|x\| & \text{otherwise.} \end{cases}$$ Show that there exists a hyperplane $H \subset \mathbb{R}^d$ such that, for all $x_0 \in \mathbb{R}^d \setminus H$, we have $\lim_{n \rightarrow \infty} f(x_n) = \min\{f(x) \mid x \in C\}$.