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 $n = 1$.

Deduce that there exist $r$ vector subspaces of $E$, denoted $E_1, \ldots, E_r$, all stable under $f$ such that:

• $E = E_1 \oplus \cdots \oplus E_r$;
• for all $1 \leqslant i \leqslant r$, the endomorphism $\psi_i$ induced by $f$ on the vector subspace $E_i$ is cyclic;
• if we denote by $P_i$ the minimal polynomial of $\psi_i$, then $P_{i+1}$ divides $P_i$ for all integer $i$ such that $1 \leqslant i \leqslant r-1$.

Deduce that there exist $r$ vector subspaces of $E$, denoted $E_1, \ldots, E_r$, all stable under $f$ such that:

• $E = E_1 \oplus \cdots \oplus E_r$;
• for all $1 \leqslant i \leqslant r$, the endomorphism $\psi_i$ induced by $f$ on the vector subspace $E_i$ is cyclic;
• if we denote by $P_i$ the minimal polynomial of $\psi_i$, then $P_{i+1}$ divides $P_i$ for all integer $i$ such that $1 \leqslant i \leqslant r-1$.

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 $E$ 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 $\mathcal{A}$.

Show that the dimension of $\mathcal{C}(f)$ is greater than or equal to $n$.

Show that the dimension of $\mathcal{C}(f)$ is greater than or equal to $n$.

Let $u$ be an endomorphism of $E$ nilpotent of index $p$ and of rank $r$.
Show that there exist a partition $\sigma = \left(\alpha_1, \ldots, \alpha_k\right)$ of $n$ and a basis $\mathcal{B}$ of $E$ in which the matrix of $u$ is equal to the matrix $N_\sigma = \operatorname{diag}\left(J_{\alpha_1}, \ldots, J_{\alpha_k}\right)$.

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 $D(u) \in \mathcal{M}_{s}(\mathbb{C})$.

We assume that $f$ is an endomorphism such that the algebra $\mathcal{C}(f)$ is equal to $\mathbb{K}[f]$. Show that $f$ is cyclic.

We assume that $f$ is an endomorphism such that the algebra $\mathcal{C}(f)$ is equal to $\mathbb{K}[f]$. Show that $f$ is cyclic.

Let $\alpha$ be a non-zero natural integer. Calculate the rank of $J_\alpha^j$ for every natural integer $j$. Deduce that $J_\alpha$ is nilpotent and specify its nilpotency index.

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.

In this part, we assume that $\mathbb{K} = \mathbb{R}$ and that $E$ is a Euclidean space. The inner product of two vectors $x, y$ of $E$ is denoted $(x \mid y)$ and we denote by $\mathrm{O}(E)$ the group of vector isometries of $E$. We say that an endomorphism $f$ of $E$ is orthocyclic if there exists an orthonormal basis of $E$ in which the matrix of $f$ is of the form $C_Q$ (companion matrix).
Let $f \in \mathrm{O}(E)$. Let $f' \in \mathrm{O}(E)$ having the same characteristic polynomial as $f$. Show that there exist orthonormal bases $\mathcal{B}$ and $\mathcal{B}'$ of $E$ for which the matrix of $f$ in $\mathcal{B}$ is equal to the matrix of $f'$ in $\mathcal{B}'$.

In this part, we assume that $\mathbb{K} = \mathbb{R}$ and that $E$ is a Euclidean space. The inner product of two vectors $x, y$ of $E$ is denoted $(x \mid y)$ and we denote by $\mathrm{O}(E)$ the group of vector isometries of $E$. We say that an endomorphism $f$ of $E$ is orthocyclic if there exists an orthonormal basis of $E$ in which the matrix of $f$ is of the form $C_Q$ (companion matrix).
Let $f \in \mathrm{O}(E)$. Let $f' \in \mathrm{O}(E)$ having the same characteristic polynomial as $f$. Show that there exist orthonormal bases $\mathcal{B}$ and $\mathcal{B}'$ of $E$ for which the matrix of $f$ in $\mathcal{B}$ is equal to the matrix of $f'$ in $\mathcal{B}'$.

Let $u$ be an endomorphism of $E$ nilpotent of index $p$ and of rank $r$. Suppose the matrix of $u$ in some basis is $N_\sigma = \operatorname{diag}\left(J_{\alpha_1}, \ldots, J_{\alpha_k}\right)$ where $\sigma = (\alpha_1, \ldots, \alpha_k)$ is a partition of $n$ with $\alpha_1 \geqslant \cdots \geqslant \alpha_k$.
Deduce the value of $\alpha_1$.

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 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 $\mathcal{A}$ is trigonalisable.

In this part, we assume that $\mathbb{K} = \mathbb{R}$ and that $E$ is a Euclidean space. We say that an endomorphism $f$ of $E$ is orthocyclic if there exists an orthonormal basis of $E$ in which the matrix of $f$ is of the form $C_Q$ (companion matrix).
Let $f \in \mathrm{O}(E)$. Deduce that $f$ is orthocyclic if and only if $\chi_f = X^n - 1$ or $\chi_f = X^n + 1$.

In this part, we assume that $\mathbb{K} = \mathbb{R}$ and that $E$ is a Euclidean space. The inner product of two vectors $x, y$ of $E$ is denoted $(x \mid y)$ and we denote by $\mathrm{O}(E)$ the group of vector isometries of $E$. We say that an endomorphism $f$ of $E$ is orthocyclic if there exists an orthonormal basis of $E$ in which the matrix of $f$ is of the form $C_Q$ (companion matrix).
Let $f \in \mathrm{O}(E)$. Deduce that $f$ is orthocyclic if and only if $\chi_f = X^n - 1$ or $\chi_f = X^n + 1$.

Let $u$ be an endomorphism of $E$ nilpotent of index $p$ and of rank $r$. Suppose the matrix of $u$ in some basis is $N_\sigma = \operatorname{diag}\left(J_{\alpha_1}, \ldots, J_{\alpha_k}\right)$ where $\sigma = (\alpha_1, \ldots, \alpha_k)$ is a partition of $n$.
For $j \in \mathbb{N}$, we denote $\Lambda_j = \left\{i \in \llbracket 1, k \rrbracket \mid \alpha_i \geqslant j\right\}$. Prove that $\operatorname{rg}\left(N_\sigma^j\right) = \sum_{i \in \Lambda_j} \left(\alpha_i - j\right)$.

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.
Show that there exists a basis of $E$ in which the matrices of elements of $\mathcal{A}$ belong to $\mathrm{T}_{n}^{+}(\mathbb{C})$.

In this part, we assume that $\mathbb{K} = \mathbb{R}$ and that $E$ is a Euclidean space. Let $f$ be a nilpotent endomorphism of $E$. Show that there exists an orthonormal basis of $E$ in which the matrix of $f$ is lower triangular.

In this part, we assume that $\mathbb{K} = \mathbb{R}$ and that $E$ is a Euclidean space. The inner product of two vectors $x, y$ of $E$ is denoted $(x \mid y)$. We say that an endomorphism $f$ of $E$ is orthocyclic if there exists an orthonormal basis of $E$ in which the matrix of $f$ is of the form $C_Q$ (companion matrix).
Let $f$ be a nilpotent endomorphism of $E$. Show that there exists an orthonormal basis of $E$ in which the matrix of $f$ is lower triangular.

Let $u$ be an endomorphism of $E$ nilpotent of index $p$ and of rank $r$. Suppose the matrix of $u$ in some basis is $N_\sigma = \operatorname{diag}\left(J_{\alpha_1}, \ldots, J_{\alpha_k}\right)$ where $\sigma = (\alpha_1, \ldots, \alpha_k)$ is a partition of $n$.
Prove that, for every $j \in \mathbb{N}^*$, the integer $d_j = \operatorname{rg}\left(u^{j-1}\right) - \operatorname{rg}\left(u^j\right)$ is equal to the number of blocks $J_{\alpha_i}$ whose size $\alpha_i$ is greater than or equal to $j$.

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 $\{ u(x) \mid u \in \mathcal{A} \}$.