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 $f$ be a nilpotent endomorphism of $E$. Deduce that $f$ is orthocyclic if and only if
$$f \text{ has rank } n-1 \quad \text{and} \quad \forall x, y \in (\ker f)^{\perp}, \quad (f(x) \mid f(y)) = (x \mid y).$$

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 $E$. Deduce that $f$ is orthocyclic if and only if
$$f \text{ has rank } n-1 \quad \text{and} \quad \forall x, y \in (\ker f)^{\perp}, \quad (f(x) \mid f(y)) = (x \mid y).$$

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 $n$.
Give the value of the integer $k$, the number of blocks $J_{\alpha_i}$ appearing in $N_\sigma$.

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 $\operatorname{Im} v$.

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 every integer $j$ between 1 and $n$, express the number of blocks $J_{\alpha_i}$ of size exactly 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 $E$).
Deduce from Q37 the existence of an element of rank 1 in $\mathcal{A}$.

Let $u$ be an endomorphism of $E$ nilpotent of index $p$ and of rank $r$. Suppose the matrix of $u$ in some basis $\mathcal{B}$ 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 $n$.
Assume that there exist a partition $\sigma'$ of the integer $n$ and a basis $\mathcal{B}'$ of $E$ such that the matrix of $u$ in $\mathcal{B}'$ is equal to $N_{\sigma'}$. Show that $\sigma = \sigma'$.

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 $i \in \llbracket 1, n \rrbracket$.

What is the maximum cardinality of a set of nilpotent matrices, all of the same size $n$, such that there are no two similar matrices in this set?

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 $A$ be the matrix $\left(\begin{array}{ccccc} 0 & -1 & 2 & -2 & -1 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 1 & -1 & 1 & 0 \end{array}\right)$ and $u$ the endomorphism canonically associated with $A$.
Determine the partition $\sigma$ of the integer 5 associated with $u$ and give the matrix $N_\sigma$.

Using the result of question 31, prove that if $M \in \mathcal{M}_n(\mathbb{C})$ is nilpotent, then $M$, $2M$ and $M^\top$ are similar.

Using the result of question 15, prove that if $M$ and $2M$ are similar, then $M$ is nilpotent.

4. Let $C_m \in \mathbb{R}[Y]$ be the characteristic polynomial of $M_m$. a. Verify that $C_1 = Y$, $C_2 = Y^2 - 2$ and $$C_m = Y C_{m-1} - m(m-1) C_{m-2}, \quad m \geq 3$$ b. Calculate the determinant of $M_m$. c. Prove that, if $e_m$ denotes the integer part of $m/2$, $$C_m = \sum_{k=0}^{e_m} (-1)^k c_{m,k} Y^{m-2k}$$ with $$c_{m,0} = 1; \quad c_{m,k} = \sum_{(a_1, \ldots, a_k) \in J_k(m)} a_1(a_1+1) a_2(a_2+1) \cdots a_k(a_k+1), \quad 1 \leq k \leq e_m;$$ where $J_k(m)$ denotes the set of $k$-tuples of integers from $\{1, \ldots, m-1\}$ such that $a_i + 2 \leq a_{i+1}$ for $1 \leq i \leq k-1$.

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.

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

Show that for all matrices $A$ and $B$ in $\operatorname{Sym}^+(p)$ and all non-negative real numbers $a$ and $b$, we have $aA + bB \in \operatorname{Sym}^+(p)$.

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 $F$ be a vector subspace of $E$ stable under $u$. Show that the orthogonal complement $F^{\perp}$ of $F$ is stable under $u$.

Let $u \in \mathcal{N}(E)$. Show that $\operatorname{tr} u^{k} = 0$ for every $k \in \mathbf{N}^{*}$.

In the case $n=1$: Show that a matrix of size $2 \times 2$ is symplectic if and only if its determinant equals 1.
Recall: $J_{1} = \left(\begin{array}{cc} 0 & 1 \\ -1 & 0 \end{array}\right)$, and a matrix $M \in \mathcal{M}_{2}(\mathbb{R})$ is symplectic if and only if $M^{\top} J_{1} M = J_{1}$.

Show that if $v \in \mathbf{R}^p$ then the matrix $A = \left(A_{ij}\right)_{(i,j) \in \llbracket 1,p \rrbracket^2}$ defined by $A = vv^T$ is in $\mathrm{Sym}^+(p)$.

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 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 $\mathcal{C}^1$.