We assume that $\mathbb{K} = \mathbb{C}$, that $(\mathrm{Id}, f, f^2, \ldots, f^{n-1})$ is free, and we factor the characteristic polynomial of $f$ in the form
$$\chi_f(X) = \prod_{k=1}^{p} \left(X - \lambda_k\right)^{m_k}$$
where the $\lambda_k$ are the $p$ eigenvalues pairwise distinct of $f$ and the $m_k \in \mathbb{N}^*$ their respective multiplicities. For $k \in \llbracket 1, p \rrbracket$, we set $F_k = \ker\left(\left(f - \lambda_k \operatorname{Id}_E\right)^{m_k}\right)$.
Show that the vector subspaces $F_k$ are stable under $f$ and that $E = F_1 \oplus \cdots \oplus F_p$.

We assume that $\mathbb{K} = \mathbb{C}$, that $(\mathrm{Id}, f, f^2, \ldots, f^{n-1})$ is free, and we factor the characteristic polynomial of $f$ in the form
$$\chi_f(X) = \prod_{k=1}^{p} \left(X - \lambda_k\right)^{m_k}$$
where the $\lambda_k$ are the $p$ eigenvalues pairwise distinct of $f$ and the $m_k \in \mathbb{N}^*$ their respective multiplicities. For $k \in \llbracket 1, p \rrbracket$, we set $F_k = \ker\left(\left(f - \lambda_k \operatorname{Id}_E\right)^{m_k}\right)$, and we denote by $\varphi_k$ the endomorphism induced by $f - \lambda_k \operatorname{Id}$ on the vector subspace $F_k$,
$$\varphi_k : \left\lvert\, \begin{aligned} & F_k \rightarrow F_k, \\ & x \mapsto f(x) - \lambda_k x. \end{aligned} \right.$$
Justify that $\varphi_k$ is a nilpotent endomorphism of $F_k$.

We assume that $\mathbb{K} = \mathbb{C}$, that $(\mathrm{Id}, f, f^2, \ldots, f^{n-1})$ is free. For $k \in \llbracket 1, p \rrbracket$, $\varphi_k$ is a nilpotent endomorphism of $F_k$, and $\nu_k$ denotes the smallest natural number such that $\varphi_k^{\nu_k} = 0$. Why do we have $\nu_k \leqslant \operatorname{dim}(F_k)$?

We assume that $\mathbb{K} = \mathbb{C}$, that $(\mathrm{Id}, f, f^2, \ldots, f^{n-1})$ is free, and we factor the characteristic polynomial of $f$ in the form
$$\chi_f(X) = \prod_{k=1}^{p} \left(X - \lambda_k\right)^{m_k}$$
where the $\lambda_k$ are the $p$ eigenvalues pairwise distinct of $f$ and the $m_k \in \mathbb{N}^*$ their respective multiplicities. For $k \in \llbracket 1, p \rrbracket$, $\varphi_k$ is a nilpotent endomorphism of $F_k$, and $\nu_k$ denotes the smallest natural number such that $\varphi_k^{\nu_k} = 0$.
Show, with the proposed hypothesis, that for all $k \in \llbracket 1, p \rrbracket$, we have $\nu_k = m_k$.

In this part, we assume $n \geqslant 2$. Let $J \in \mathcal{M}_{n}(\mathbb{R})$ be the matrix canonically associated with the endomorphism $\varphi \in \mathcal{L}(\mathbb{R}^{n})$ defined by $\varphi: e_{j} \mapsto e_{j+1}$ if $j \in \{1, \ldots, n-1\}$ and $\varphi(e_{n}) = e_{1}$, where $(e_{1}, \ldots, e_{n})$ is the canonical basis of $\mathbb{R}^{n}$.
Show that $J$ is diagonalisable in $\mathcal{M}_{n}(\mathbb{C})$.

In this part, we assume $n \geqslant 2$. Let $J \in \mathcal{M}_{n}(\mathbb{R})$ be the matrix canonically associated with the endomorphism $\varphi \in \mathcal{L}(\mathbb{R}^{n})$ defined by $\varphi: e_{j} \mapsto e_{j+1}$ if $j \in \{1, \ldots, n-1\}$ and $\varphi(e_{n}) = e_{1}$, where $(e_{1}, \ldots, e_{n})$ is the canonical basis of $\mathbb{R}^{n}$.
Is the matrix $J$ diagonalisable in $\mathcal{M}_{n}(\mathbb{R})$?

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 $A$.
Prove that $A$ is similar to the matrix $\operatorname{diag}\left(J_2, J_1\right)$. Give the value of an invertible matrix $P$ such that $A = P \operatorname{diag}\left(J_2, J_1\right) P^{-1}$.

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 $R' = P^{-1}RP$.

We propose to study the matrix equation $R^2 = J_3$.
Let $R$ be a solution of this equation. Give the values of $R^4$ and $R^6$, then the set of solutions of the equation.

Let $V \in \mathcal{M}_n(\mathbb{C})$ be a nilpotent matrix of index $p$. We propose to study the equation $R^2 = V$.
Show that, if $2p - 1 > n$, then there is no solution.

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 denote by $d$ the degree of $\pi_f$, $E_1 = \operatorname{Vect}(x_1, f(x_1), \ldots, f^{d-1}(x_1))$, and $\psi_1$ is the endomorphism induced by $f$ on the vector subspace $E_1$,
$$\psi_1 : \left\lvert\, \begin{aligned} & E_1 \rightarrow E_1, \\ & x \mapsto f(x). \end{aligned} \right.$$
Justify that $\psi_1$ is cyclic.

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 $\operatorname{Im}(u)$.

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 $N_\sigma$.

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

The purpose of this question is to show that $\sqrt { 3 }$ is not an eigenvalue of a matrix in $S _ { 2 } ( \mathbb { Q } )$. We assume that there exists $M \in S _ { 2 } ( \mathbb { Q } )$ such that $\sqrt { 3 }$ is an eigenvalue of $M$.
2a. Using the irrationality of $\sqrt { 3 }$, show that the characteristic polynomial of $M$ is $X ^ { 2 } - 3$.
2b. Show that if $n \in \mathbb { Z }$, then $n ^ { 2 }$ is congruent to 0 or 1 modulo 3.
2c. Show that there does not exist a triple of integers $(x, y, z)$ that are coprime as a whole such that $x ^ { 2 } + y ^ { 2 } = 3 z ^ { 2 }$.
2d. Conclude.

The purpose of this question is to show that $\sqrt [ 3 ] { 2 }$ is not an eigenvalue of a symmetric matrix with coefficients in $\mathbb { Q }$. We reason by contradiction, assuming the existence of a matrix $M \in S _ { n } ( \mathbb { Q } )$ (for some integer $n$) for which $\sqrt [ 3 ] { 2 }$ is an eigenvalue.
4a. Show that $X ^ { 3 } - 2$ divides the characteristic polynomial of $M$. (One may begin by proving that $\sqrt [ 3 ] { 2 } \notin \mathbb { Q }$.)
4b. Conclude.

In the case $n=1$: Let $M$ be a matrix of size $2 \times 2$ that is symmetric and symplectic. Show that $M$ is diagonalizable and that its eigenvalues are inverses of each other. Show that there exists a matrix $P$ that is both orthogonal and symplectic such that $P^{-1} M P$ is diagonal.

For $n \in \mathbb { N } ^ { * }$, construct a matrix $M \in S _ { n } ( \mathbb { Q } )$ for which $\cos \left( \frac { 2 \pi } { n } \right)$ is an eigenvalue. (One may begin by constructing an orthogonal matrix with coefficients in $\mathbb { Q }$ that admits $e ^ { 2 i \pi / n }$ as an eigenvalue.)

For $n \in \mathbb { N } ^ { \star }$, construct a matrix $M \in S _ { n } ( \mathbb { Q } )$ for which $\cos \left( \frac { 2 \pi } { n } \right)$ is an eigenvalue. (One may begin by constructing an orthogonal matrix with coefficients in $\mathbb { Q }$ that admits $e ^ { 2 i \pi / n }$ as an eigenvalue.)

In the case $n=1$: Determine the matrices of size $2 \times 2$ that are both antisymmetric and symplectic and show that they are not diagonalizable in $\mathbb{R}$.