Let $E$ be a $\mathbb{C}$-vector space of dimension $n \geq 1$. We say that an endomorphism $u$ of $E$ is a permutation endomorphism if there exists a basis $(e_1, \ldots, e_n)$ of $E$ and a permutation $\sigma \in \mathfrak{S}_n$ such that $u(e_j) = e_{\sigma(j)}$ for all $j \in \llbracket 1, n \rrbracket$.
Show that $u$ is a permutation endomorphism if and only if there exists a basis in which its matrix is a permutation matrix.

Let $M \in \mathcal{A}_{2n}(\mathbb{R}) \cap \mathrm{Sp}_{2n}(\mathbb{R})$. Show that there exists $P \in \mathcal{O}_{2n}(\mathbb{R}) \cap \mathrm{Sp}_{2n}(\mathbb{R})$ such that $P^{\top} M^{2} P$ is diagonal with diagonal coefficients $d_{1}, \ldots, d_{2n}$ satisfying for all $k \in \{1, \ldots, n\}$, $d_{k+n} = 1/d_{k}$.

In this subsection, $E$ is a $\mathbb{C}$-vector space of dimension $n \geq 1$. We say that an endomorphism $u$ of $E$ is a permutation endomorphism if there exists a basis $(e_1, \ldots, e_n)$ of $E$ and a permutation $\sigma \in \mathfrak{S}_n$ such that $u(e_j) = e_{\sigma(j)}$ for all $j \in \llbracket 1, n \rrbracket$.
Show that $u$ is a permutation endomorphism if and only if there exists a basis in which its matrix is a permutation matrix.

Let $E$ be a $\mathbb{C}$-vector space of dimension $n \geq 1$. Let $u$ be a permutation endomorphism of $E$. Show that $u$ is diagonalizable and that its trace belongs to $\llbracket 0, n \rrbracket$.

Let $M \in \mathcal{A}_{2n}(\mathbb{R}) \cap \mathrm{Sp}_{2n}(\mathbb{R})$, and let $m$ be the linear map canonically associated with $M$. Let $X$ denote an eigenvector of $M^{2}$ of norm 1 associated with a certain eigenvalue $\lambda$. Show that $MX$, $J_{n} X$ and $J_{n} MX$ are eigenvectors of $M^{2}$ and give the eigenvalues associated with each of these vectors.

In this subsection, $E$ is a $\mathbb{C}$-vector space of dimension $n \geq 1$. We say that an endomorphism $u$ of $E$ is a permutation endomorphism if there exists a basis $(e_1, \ldots, e_n)$ of $E$ and a permutation $\sigma \in \mathfrak{S}_n$ such that $u(e_j) = e_{\sigma(j)}$ for all $j \in \llbracket 1, n \rrbracket$.
Let $u$ be a permutation endomorphism of $E$. Show that $u$ is diagonalizable and that its trace belongs to $\llbracket 0, n \rrbracket$.

Let $A, B$ be two diagonalizable matrices of $\mathcal{M}_n(\mathbb{C})$. Show that $A$ and $B$ are similar if and only if they have the same characteristic polynomial.

Let $M \in \mathcal{A}_{2n}(\mathbb{R}) \cap \mathrm{Sp}_{2n}(\mathbb{R})$, and let $m$ be the linear map canonically associated with $M$. Let $X$ denote an eigenvector of $M^{2}$ of norm 1 associated with a certain eigenvalue $\lambda$. Denote $F = \operatorname{Vect}(X, MX, J_{n} X, J_{n} MX)$. Show that $F$ is stable under $M$ and under $J_{n}$.

Let $A, B$ be two diagonalizable matrices of $\mathcal{M}_n(\mathbb{C})$. Show that $A$ and $B$ are similar if and only if they have the same characteristic polynomial.

Let $E$ be a $\mathbb{C}$-vector space of dimension $n \geq 1$. Let $u$ be an endomorphism of $E$ such that $u^2 = \operatorname{Id}_E$. Show that $u$ is a permutation endomorphism if and only if $\operatorname{Tr}(u)$ is a natural integer.

Throughout part III, $N$ is a fixed non-zero natural integer and $(X_n)_{n \in \mathbb{N}}$ is a homogeneous Markov chain on $\llbracket 0, N \rrbracket$ with transition matrix $Q$ where $q_{i,j} = P(X_{n+1} = j \mid X_n = i) > 0$. We define $a_{i,j}(t) = q_{i,j} \mathrm{e}^{jt}$ and $A(t) = \left(a_{i,j}(t)\right)_{0 \leqslant i,j \leqslant N} \in \mathcal{M}_{N+1}(\mathbb{R})$. Justify that $A(t)$ possesses a dominant eigenvalue $\gamma(t) > 0$.

Let $M \in \mathcal{A}_{2n}(\mathbb{R}) \cap \mathrm{Sp}_{2n}(\mathbb{R})$. Show that all eigenvalues of $M^{2}$ are strictly negative.

In this subsection, $E$ is a $\mathbb{C}$-vector space of dimension $n \geq 1$. We say that an endomorphism $u$ of $E$ is a permutation endomorphism if there exists a basis $(e_1, \ldots, e_n)$ of $E$ and a permutation $\sigma \in \mathfrak{S}_n$ such that $u(e_j) = e_{\sigma(j)}$ for all $j \in \llbracket 1, n \rrbracket$.
Let $u$ be an endomorphism of $E$ such that $u^2 = \operatorname{Id}_E$. Show that $u$ is a permutation endomorphism if and only if $\operatorname{Tr}(u)$ is a natural integer.

Let $E$ be a $\mathbb{C}$-vector space of dimension $n \geq 1$. Study whether the equivalence of the previous question holds when we replace the hypothesis $u^2 = \operatorname{Id}_E$ by $u^k = \operatorname{Id}_E$ for $k = 3$, then for $k = 4$.

Throughout part III, $N$ is a fixed non-zero natural integer and $(X_n)_{n \in \mathbb{N}}$ is a homogeneous Markov chain on $\llbracket 0, N \rrbracket$ with transition matrix $Q$ where $q_{i,j} = P(X_{n+1} = j \mid X_n = i) > 0$. We define $a_{i,j}(t) = q_{i,j} \mathrm{e}^{jt}$, $A(t) = \left(a_{i,j}(t)\right)_{0 \leqslant i,j \leqslant N}$, $z_j(t) = P(X_1 = j)\mathrm{e}^{jt}$, $Z(t) = \begin{pmatrix} z_0(t) \\ \vdots \\ z_N(t) \end{pmatrix}$, and $Y^{(n)}(t) = (A(t))^{n-1} Z(t)$ so that $E\left(\mathrm{e}^{tS_n}\right) = \sum_{j=0}^N Y_j^{(n)}(t)$. Let $\gamma(t)$ be the dominant eigenvalue of $A(t)$. Show that $\lim_{n \rightarrow +\infty} \frac{\ln\left(E\left(\mathrm{e}^{tS_n}\right)\right)}{n} = \lambda(t)$ where $\lambda(t) = \ln(\gamma(t))$.

Let $M \in \mathcal{A}_{2n}(\mathbb{R}) \cap \mathrm{Sp}_{2n}(\mathbb{R})$, and let $m$ be the linear map canonically associated with $M$. Let $X$ denote an eigenvector of $M^{2}$ of norm 1 associated with a certain eigenvalue $\lambda$, and let $F = \operatorname{Vect}(X, MX, J_{n} X, J_{n} MX)$. Justify that if $\lambda \neq -1$, $F$ is a vector space of dimension 4. Show that, in this case, $$\left(X,\ \frac{-1}{\sqrt{-\lambda}} MX,\ -J_{n} X,\ \frac{1}{\sqrt{-\lambda}} J_{n} MX\right)$$ is an orthonormal basis of $F$. Then give the matrix of the application $m_{F}$ induced by $m$ on $F$ in the basis obtained.

In this subsection, $E$ is a $\mathbb{C}$-vector space of dimension $n \geq 1$. We say that an endomorphism $u$ of $E$ is a permutation endomorphism if there exists a basis $(e_1, \ldots, e_n)$ of $E$ and a permutation $\sigma \in \mathfrak{S}_n$ such that $u(e_j) = e_{\sigma(j)}$ for all $j \in \llbracket 1, n \rrbracket$.
Study whether the equivalence of the previous question holds when we replace the hypothesis $u^2 = \operatorname{Id}_E$ by $u^k = \operatorname{Id}_E$ for $k = 3$, then for $k = 4$.

Let $E$ be a $\mathbb{C}$-vector space of dimension $n \geq 1$. Let $u$ be an endomorphism of $E$. Show that $u$ is a permutation endomorphism if and only if it satisfies the following two conditions:
(a) there exist natural integers $c_1, \ldots, c_n$ such that $\chi_u = \prod_{\ell=1}^{n} \left(X^\ell - 1\right)^{c_\ell}$;
(b) there exists $N$ such that $u^N = \operatorname{Id}_E$.

Write in Python language a function \texttt{puiss2k} that takes as argument a square matrix $M$ and a natural integer $k$ and returns the matrix $M^{2^k}$ by performing $k$ matrix products. One may exploit the fact that $M^{2^{k+1}} = M^{2^k} M^{2^k}$.

Let $M \in \mathcal{A}_{2n}(\mathbb{R}) \cap \mathrm{Sp}_{2n}(\mathbb{R})$, and let $m$ be the linear map canonically associated with $M$. Let $X$ denote an eigenvector of $M^{2}$ of norm 1 associated with a certain eigenvalue $\lambda$, and let $F = \operatorname{Vect}(X, MX, J_{n} X, J_{n} MX)$. Show that $F^{\perp}$ is stable under $M$ and under $J_{n}$.

In this subsection, $E$ is a $\mathbb{C}$-vector space of dimension $n \geq 1$. We say that an endomorphism $u$ of $E$ is a permutation endomorphism if there exists a basis $(e_1, \ldots, e_n)$ of $E$ and a permutation $\sigma \in \mathfrak{S}_n$ such that $u(e_j) = e_{\sigma(j)}$ for all $j \in \llbracket 1, n \rrbracket$.
Let $u$ be an endomorphism of $E$. Show that $u$ is a permutation endomorphism if and only if it satisfies the following two conditions:
(a) there exist natural integers $c_1, \ldots, c_n$ such that $\chi_u = \prod_{\ell=1}^{n} \left(X^\ell - 1\right)^{c_\ell}$;
(b) there exists $N$ such that $u^N = \operatorname{Id}_E$.

Let $E$ be a $\mathbb{C}$-vector space of dimension $n \geq 1$. Let $u$ and $v$ be two endomorphisms of $E$ such that, for all $k \in \mathbb{N}, \operatorname{Tr}\left(u^k\right) = \operatorname{Tr}\left(v^k\right)$. Show that $u$ and $v$ have the same characteristic polynomial.

Explain what the Python function \texttt{maxSp} defined by: \begin{verbatim} def maxSp(Q:np.ndarray, k:int, t:float) -> float: n = Q.shape[1] E = np.exp(t * np.array(range(n))) A = Q * E B = puiss2k(A, k) C = np.dot(A, B) return C.trace() / B.trace() \end{verbatim} does.

Let $M \in \mathcal{A}_{2n}(\mathbb{R}) \cap \mathrm{Sp}_{2n}(\mathbb{R})$, and let $m$ be the linear map canonically associated with $M$. Show that there exists a non-zero natural integer $q$ and vector subspaces of $\mathcal{M}_{2n,1}(\mathbb{R})$, denoted $F_{1}, \ldots, F_{q}$ such that

• [(a)] $F_{1} \oplus \cdots \oplus F_{q} = \mathcal{M}_{2n,1}(\mathbb{R})$;
• [(b)] $\forall i \in \{1,\ldots,q\}$, $F_{i}$ is stable under $M$ and under $J_{n}$;
• [(c)] $\forall i \in \{1,\ldots,q\}$, $F_{i}^{\perp}$ is stable under $M$ and under $J_{n}$;
• [(d)] $\forall (i,j) \in \{1,\ldots,q\}^{2}$, $i \neq j \Longrightarrow \forall (Y,Z) \in F_{i} \times F_{j}$, $\langle Y,Z \rangle = 0 = \varphi(Y,Z)$;
• [(e)] $\forall i \in \{1,\ldots,q\}$, $\dim F_{i} \in \{2,4\}$;
• [(f)] $\forall i \in \{1,\ldots,q\}$, the matrix of the application $m_{F_{i}}$ induced by $m$ on $F_{i}$ in a certain basis is of the form $$J_{1} \quad \text{or} \quad \left(\begin{array}{cc} \sqrt{-\lambda} J_{1} & 0_{2,2} \\ 0_{2,2} & \frac{1}{\sqrt{-\lambda}} J_{1} \end{array}\right).$$

Let $u$ and $v$ be two endomorphisms of $E$ such that, for all $k \in \mathbb{N}, \operatorname{Tr}\left(u^k\right) = \operatorname{Tr}\left(v^k\right)$. Show that $u$ and $v$ have the same characteristic polynomial.