We assume that $n = 2$. Let $u$ be an endomorphism of $E$ nilpotent of index $p \geqslant 2$.
Show that there exists a vector $x$ in $E$ such that $u^{p-1}(x) \neq 0$.

We assume that $n = 2$. Let $u$ be an endomorphism of $E$ nilpotent of index $p \geqslant 2$.
Verify that the family $\left(u^{k}(x)\right)_{0 \leqslant k \leqslant p-1}$ is free. Deduce that $p = 2$.

We assume that $n = 2$. Let $u$ be an endomorphism of $E$ nilpotent of index $p \geqslant 2$.
Show that $\operatorname{Ker} u = \operatorname{Im} u$.

We assume that $n = 2$. Let $u$ be an endomorphism of $E$ nilpotent of index $p \geqslant 2$.
Construct a basis of $E$ in which the matrix of $u$ is equal to $J_2$.

We assume that $n \geqslant 3$. Let $u$ be an endomorphism of $E$ nilpotent of index 2 and of rank $r$.
Show that $\operatorname{Im} u \subset \operatorname{Ker} u$ and that $2r \leqslant n$.

We assume that $n \geqslant 3$. Let $u$ be an endomorphism of $E$ nilpotent of index 2 and of rank $r$.
Assume that $\operatorname{Im} u = \operatorname{Ker} u$. Show that there exist vectors $e_1, e_2, \ldots, e_r$ in $E$ such that $\left(e_1, u\left(e_1\right), e_2, u\left(e_2\right), \ldots, e_r, u\left(e_r\right)\right)$ is a basis of $E$.

We assume that $n \geqslant 3$. Let $u$ be an endomorphism of $E$ nilpotent of index 2 and of rank $r$. Assume that $\operatorname{Im} u = \operatorname{Ker} u$ and that $\left(e_1, u\left(e_1\right), e_2, u\left(e_2\right), \ldots, e_r, u\left(e_r\right)\right)$ is a basis of $E$.
Give the matrix of $u$ in this basis.

We assume that $n \geqslant 3$. Let $u$ be an endomorphism of $E$ nilpotent of index 2 and of rank $r$.
Assume $\operatorname{Im} u \neq \operatorname{Ker} u$. Show that there exist vectors $e_1, e_2, \ldots, e_r$ in $E$ and vectors $v_1, v_2, \ldots, v_{n-2r}$ belonging to $\operatorname{Ker} u$ such that $\left(e_1, u\left(e_1\right), e_2, u\left(e_2\right), \ldots, e_r, u\left(e_r\right), v_1, \ldots, v_{n-2r}\right)$ is a basis of $E$.

We assume that $n \geqslant 3$. Let $u$ be an endomorphism of $E$ nilpotent of index 2 and of rank $r$. Assume $\operatorname{Im} u \neq \operatorname{Ker} u$ and that $\left(e_1, u\left(e_1\right), e_2, u\left(e_2\right), \ldots, e_r, u\left(e_r\right), v_1, \ldots, v_{n-2r}\right)$ is a basis of $E$.
What is the matrix of $u$ in this basis?

Let $A$ denote a matrix in $\mathcal{M}_n(\mathbb{C})$.
Prove that, if $A$ is a nilpotent matrix of index $p$, then every polynomial in $\mathbb{C}[X]$ that is a multiple of $X^p$ is an annihilating polynomial of $A$.

We denote $A = \left(\begin{array}{ccc} 1 & 3 & -7 \\ 2 & 6 & -14 \\ 1 & 3 & -7 \end{array}\right)$ and $u$ the endomorphism of $\mathbb{C}^3$ canonically associated with $A$. We seek to determine the set of matrices $R \in \mathcal{M}_3(\mathbb{C})$ such that $R^2 = A$. We denote by $\rho$ the endomorphism canonically associated with $R$.
Prove that $\operatorname{Im} u$ and $\operatorname{Ker} u$ are stable under $\rho$ and that $\rho$ is nilpotent.

We denote $A = \left(\begin{array}{ccc} 1 & 3 & -7 \\ 2 & 6 & -14 \\ 1 & 3 & -7 \end{array}\right)$ and $u$ the endomorphism of $\mathbb{C}^3$ canonically associated with $A$. We seek to determine the set of matrices $R \in \mathcal{M}_3(\mathbb{C})$ such that $R^2 = A$. We denote by $\rho$ the endomorphism canonically associated with $R$.
Deduce the set of square roots of $A$. One may consider $R' = P^{-1}RP$.

Let $V \in \mathcal{M}_n(\mathbb{C})$ be a nilpotent matrix of index $p$. We propose to study the equation $R^2 = V$.
For every value of the integer $n \geqslant 3$, exhibit a matrix $V \in \mathcal{M}_n(\mathbb{C})$, nilpotent of index $p \geqslant 2$ and admitting at least one square root.

We assume $n \geqslant 2$. Let $u$ be an endomorphism of $E$ nilpotent of index $p \geqslant 2$.
Prove that $\operatorname{Im} u$ is stable under $u$ and that the endomorphism induced by $u$ on $\operatorname{Im} u$ is nilpotent. Specify its nilpotency index.

We assume $n \geqslant 2$. Let $u$ be an endomorphism of $E$ nilpotent of index $p \geqslant 2$.
For every non-zero vector $x$ in $E$, we denote by $C_u(x)$ the vector space spanned by the $\left(u^k(x)\right)_{k \in \mathbb{N}}$; prove that $C_u(x)$ is stable under $u$ and that there exists a smallest integer $s(x) \geqslant 1$ such that $u^{s(x)}(x) = 0$.

We assume $n \geqslant 2$. Let $u$ be an endomorphism of $E$ nilpotent of index $p \geqslant 2$. For every non-zero vector $x$ in $E$, we denote by $C_u(x)$ the vector space spanned by the $\left(u^k(x)\right)_{k \in \mathbb{N}}$ and $s(x)$ the smallest integer $\geqslant 1$ such that $u^{s(x)}(x) = 0$.
Prove that $(x, u(x), \ldots, u^{s(x)-1}(x))$ is a basis of $C_u(x)$ and give the matrix, in this basis, of the endomorphism induced by $u$ on $C_u(x)$.

We assume $n \geqslant 2$. Let $u$ be an endomorphism of $E$ nilpotent of index $p \geqslant 2$.
Prove by induction on $p$ that there exist vectors $x_1, \ldots, x_t$ in $E$ such that $E = \bigoplus_{i=1}^{t} C_u\left(x_i\right)$. One may apply the induction hypothesis to the endomorphism induced by $u$ on $\operatorname{Im}(u)$.

We assume $n \geqslant 2$. Let $u$ be an endomorphism of $E$ nilpotent of index $p \geqslant 2$, and suppose there exist vectors $x_1, \ldots, x_t$ in $E$ such that $E = \bigoplus_{i=1}^{t} C_u\left(x_i\right)$.
Give the matrix of $u$ in a basis adapted to the decomposition $E = \bigoplus_{i=1}^{t} C_u\left(x_i\right)$.

Let $F$ be a vector subspace of $E$ of dimension $p$ and $\mathcal{A}_{F}$ the set of endomorphisms of $E$ that stabilise $F$, that is $\mathcal{A}_{F} = \{ u \in \mathcal{L}(E) \mid u(F) \subset F \}$.
Show that $\mathcal{A}_{F}$ is a subalgebra of $\mathcal{L}(E)$.

Throughout this part, $\mathcal{A}$ is a subalgebra of $\mathcal{M}_{n}(\mathbb{R})$ strictly contained in $\mathcal{M}_{n}(\mathbb{R})$ and we denote by $d$ its dimension. We denote by $\mathcal{A}^{\perp}$ the orthogonal complement of $\mathcal{A}$ in $\mathcal{M}_{n}(\mathbb{R})$ and we denote by $r$ its dimension. We fix a basis $(A_{1}, \ldots, A_{r})$ of $\mathcal{A}^{\perp}$. Let $\mathcal{A}^{\top} = \{ M^{\top} \mid M \in \mathcal{A} \}$. We denote by $\mathcal{M}_{n,1}(\mathbb{R})$ the $\mathbb{R}$-vector space of column matrices with $n$ rows and real coefficients.
Let $X \in \mathcal{M}_{n,1}(\mathbb{R})$ and let $F = \operatorname{Vect}(A_{1}X, \ldots, A_{r}X)$. Show that $F$ is stable by the endomorphisms of $\mathcal{M}_{n,1}(\mathbb{R})$ canonically associated with elements of $\mathcal{A}^{\top}$.

Let $E$ be a $\mathbb{C}$-vector space of finite dimension $n \geqslant 1$. Let $\mathcal{A}$ be a subalgebra of $\mathcal{L}(E)$ consisting of nilpotent endomorphisms. We propose to prove by strong induction on $n \in \mathbb{N}^{*}$ that if all elements of $\mathcal{A}$ are nilpotent, then $\mathcal{A}$ is trigonalisable.
Show that the result is true if $n = 1$.

Let $E$ be a $\mathbb{C}$-vector space of finite dimension $n \geqslant 1$. Let $\mathcal{A}$ be a subalgebra of $\mathcal{L}(E)$ consisting of nilpotent endomorphisms. We assume that $n \geqslant 2$ and that the result (that $\mathcal{A}$ is trigonalisable) is true for all natural integers $d \leqslant n-1$.
We admit Burnside's Theorem: Let $E$ be a $\mathbb{C}$-vector space of dimension $n \geqslant 2$. Let $\mathcal{A}$ be a subalgebra of $\mathcal{L}(E)$. If the only vector subspaces of $E$ stable by all elements of $\mathcal{A}$ are $\{0\}$ and $E$, then $\mathcal{A} = \mathcal{L}(E)$.
Show that there exists a vector subspace $V$ of $E$ distinct from $E$ and $\{0\}$ stable by all elements of $\mathcal{A}$.

Let $E$ be a $\mathbb{C}$-vector space of finite dimension $n \geqslant 1$. Let $\mathcal{A}$ be a subalgebra of $\mathcal{L}(E)$ consisting of nilpotent endomorphisms. We assume that $n \geqslant 2$ and that the result is true for all natural integers $d \leqslant n-1$. We fix a vector subspace $V$ of $E$ distinct from $E$ and $\{0\}$ stable by all elements of $\mathcal{A}$, and denote by $r$ its dimension. Let also $s = n - r$.
Show that there exists a basis $\mathcal{B}$ of $E$ such that for all $u \in \mathcal{A}$, $$\operatorname{Mat}_{\mathcal{B}}(u) = \left( \begin{array}{cc} A(u) & B(u) \\ 0 & D(u) \end{array} \right)$$ where $A(u) \in \mathcal{M}_{r}(\mathbb{C})$, $B(u) \in \mathcal{M}_{r,s}(\mathbb{C})$ and $D(u) \in \mathcal{M}_{s}(\mathbb{C})$.

Let $E$ be a $\mathbb{C}$-vector space of finite dimension $n \geqslant 1$. Let $\mathcal{A}$ be a subalgebra of $\mathcal{L}(E)$ consisting of nilpotent endomorphisms. We assume that $n \geqslant 2$ and that the result is true for all natural integers $d \leqslant n-1$. We fix a vector subspace $V$ of $E$ distinct from $E$ and $\{0\}$ stable by all elements of $\mathcal{A}$, and denote by $r$ its dimension. Let also $s = n - r$. There exists a basis $\mathcal{B}$ of $E$ such that for all $u \in \mathcal{A}$, $$\operatorname{Mat}_{\mathcal{B}}(u) = \left( \begin{array}{cc} A(u) & B(u) \\ 0 & D(u) \end{array} \right)$$ where $A(u) \in \mathcal{M}_{r}(\mathbb{C})$, $B(u) \in \mathcal{M}_{r,s}(\mathbb{C})$ and $D(u) \in \mathcal{M}_{s}(\mathbb{C})$.
Show that $\{ A(u) \mid u \in \mathcal{A} \}$ is a subalgebra of $\mathcal{M}_{r}(\mathbb{C})$ consisting of nilpotent matrices and that $\{ D(u) \mid u \in \mathcal{A} \}$ is a subalgebra of $\mathcal{M}_{s}(\mathbb{C})$ consisting of nilpotent matrices.

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 $x$ and $y$ be two elements of $E$, with $x$ being non-zero. Show that there exists $u \in \mathcal{A}$ such that $u(x) = y$.
One may consider in $E$ the vector subspace $\{ u(x) \mid u \in \mathcal{A} \}$.