We call the commutant of $f$ the set $\mathcal{C}(f) = \{g \in \mathcal{L}(E) \mid f \circ g = g \circ f\}$. Show that $\mathcal{C}(f)$ is a subalgebra of $\mathcal{L}(E)$.

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 $A$.
Calculate the trace and rank of $A$. Deduce, without any calculation, the characteristic polynomial of $A$. Show that $A$ is nilpotent and give its nilpotency index.

In this part, we assume $n \geqslant 2$. For all $(a_{0}, \ldots, a_{n-1}) \in \mathbb{R}^{n}$, we set $$J(a_{0}, \ldots, a_{n-1}) = \left( \begin{array}{cccc} a_{0} & a_{n-1} & \cdots & a_{1} \\ a_{1} & a_{0} & \cdots & a_{2} \\ \vdots & \vdots & & \vdots \\ a_{n-1} & a_{n-2} & \cdots & a_{0} \end{array} \right)$$ Let $\mathcal{A}$ be the set of matrices of $\mathcal{M}_{n}(\mathbb{R})$ of the form $J(a_{0}, \ldots, a_{n-1})$ where $(a_{0}, \ldots, a_{n-1}) \in \mathbb{R}^{n}$.
Is the subset $\mathcal{A}$ a subalgebra of $\mathcal{M}_{n}(\mathbb{C})$?

In this part, we assume $n \geqslant 2$. For all $(a_{0}, \ldots, a_{n-1}) \in \mathbb{R}^{n}$, we set $$J(a_{0}, \ldots, a_{n-1}) = \left( \begin{array}{cccc} a_{0} & a_{n-1} & \cdots & a_{1} \\ a_{1} & a_{0} & \cdots & a_{2} \\ \vdots & \vdots & & \vdots \\ a_{n-1} & a_{n-2} & \cdots & a_{0} \end{array} \right)$$ Let $\mathcal{A}$ be the set of matrices of $\mathcal{M}_{n}(\mathbb{R})$ of the form $J(a_{0}, \ldots, a_{n-1})$ where $(a_{0}, \ldots, a_{n-1}) \in \mathbb{R}^{n}$.
Show that there exists $P \in \mathrm{GL}_{n}(\mathbb{C})$ such that, for every matrix $A \in \mathcal{A}$, the matrix $P^{-1}AP$ is diagonal.

In this part, we assume $n \geqslant 2$. For all $(a_{0}, \ldots, a_{n-1}) \in \mathbb{R}^{n}$, we set $$J(a_{0}, \ldots, a_{n-1}) = \left( \begin{array}{cccc} a_{0} & a_{n-1} & \cdots & a_{1} \\ a_{1} & a_{0} & \cdots & a_{2} \\ \vdots & \vdots & & \vdots \\ a_{n-1} & a_{n-2} & \cdots & a_{0} \end{array} \right)$$ Let $(a_{0}, \ldots, a_{n-1}) \in \mathbb{R}^{n}$. We denote by $Q \in \mathbb{R}[X]$ the polynomial $\sum_{k=0}^{n-1} a_{k} X^{k}$.
What are the complex eigenvalues of the matrix $J(a_{0}, \ldots, a_{n-1})$?

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. The trace of any matrix $M$ of $\mathcal{M}_{n}(\mathbb{R})$ is denoted $\operatorname{tr}(M)$.
Show that the map defined on $\mathcal{M}_{n}(\mathbb{R}) \times \mathcal{M}_{n}(\mathbb{R})$ by $(A, B) \mapsto \langle A \mid B \rangle = \operatorname{tr}(A^{\top} B)$ is an inner product on $\mathcal{M}_{n}(\mathbb{R})$.

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 $\mathcal{M}_{n}(\mathbb{R})$ (with respect to the inner product $(A,B) \mapsto \operatorname{tr}(A^\top B)$) and we denote by $r$ its dimension.
What relationship holds between $d$ and $r$?

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 $M \in \mathcal{M}_{n}(\mathbb{R})$. Show that $M$ belongs to $\mathcal{A}$ if and only if, for all $i \in \llbracket 1, r \rrbracket$, $\langle A_{i} \mid M \rangle = 0$.

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 $\mathcal{A}^{\perp}$.
Show that for every matrix $N \in \mathcal{A}$ and all $i \in \llbracket 1, r \rrbracket$, we have $N^{\top} A_{i} \in \mathcal{A}^{\perp}$.

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. Let $\mathcal{A}^{\top} = \{ M^{\top} \mid M \in \mathcal{A} \}$.
Show that $\mathcal{A}^{\top}$ is a subalgebra of $\mathcal{M}_{n}(\mathbb{R})$ of the same dimension as $\mathcal{A}$.

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 $\mathcal{A}^{\top}$.

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 $\mathcal{A}^{\perp}$.
Show that $d \leqslant n^{2} - n + 1$ and conclude.

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.

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.

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 $\mathrm{T}_{n}^{+}(\mathbb{C})$.

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

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 $v \in \mathcal{A}$ of rank greater than or equal to 2. Show that there exists $u \in \mathcal{A}$ and $\lambda \in \mathbb{C}$ such that $$0 < \operatorname{rg}(v \circ u \circ v - \lambda v) < \operatorname{rg} v.$$ Consider $x$ and $y$ in $E$ such that the family $(v(x), v(y))$ is free, justify the existence of $u \in \mathcal{A}$ such that $u \circ v(x) = y$ and consider the endomorphism induced by $v \circ u$ on $\operatorname{Im} v$.

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 $E$).
Deduce from Q37 the existence of an element of rank 1 in $\mathcal{A}$.

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 $\mathcal{A} = \mathcal{L}(E)$).

In this question only, $n$ is any non-zero natural integer. Determine $J_{n}^{2}$ and show that $J_{n} \in \mathrm{Sp}_{2n}(\mathbb{R}) \cap \mathcal{A}_{2n}(\mathbb{R})$.
Recall: $J_{n} = \left(\begin{array}{cc} 0_{n,n} & I_{n} \\ -I_{n} & 0_{n,n} \end{array}\right)$, and a matrix $M \in \mathcal{M}_{2n}(\mathbb{R})$ is symplectic if and only if $M^{\top} J_{n} M = J_{n}$.

Exhibit a matrix $M \in S _ { 2 } ( \mathbb { Q } )$ for which $\sqrt { 2 }$ is an eigenvalue.