We say that a complex number $z$ is totally real (resp. totally positive) if there exists a non-zero polynomial $P ( X )$ with rational coefficients such that: (i) $z$ is a root of $P$, and (ii) all roots of $P$ are in $\mathbb { R }$ (resp. in $\mathbb { R } _ { + }$).
Let $M$ be a symmetric matrix with coefficients in $\mathbb { Q }$. Show that the eigenvalues of $M$ are totally real.

We say that a complex number $z$ is totally real (resp. totally positive) if there exists a non-zero polynomial $P ( X )$ with rational coefficients such that: (i) $z$ is a root of $P$, and (ii) all roots of $P$ are in $\mathbb { R }$ (resp. in $\mathbb { R } _ { + }$).
Let $M$ be a symmetric matrix with coefficients in $\mathbb { Q }$. Show that the eigenvalues of $M$ are totally real.

Let $M \in \mathcal{S}_{2n}(\mathbb{R}) \cap \mathrm{Sp}_{2n}(\mathbb{R})$. Show that if $\lambda$ is an eigenvalue of $M$, then $1/\lambda$ is also an eigenvalue of $M$. Give an eigenvector associated with it.

By reasoning on the multiplicity of the roots of $\chi_\sigma$ and $\chi_\tau$, show that if $P_\sigma$ and $P_\tau$ are similar, then, for all $q \in \llbracket 1, n \rrbracket$,
$$\sum_{\substack{\ell=1 \\ q \mid \ell}}^{n} c_\ell(\sigma) = \sum_{\substack{\ell=1 \\ q \mid \ell}}^{n} c_\ell(\tau)$$
(We sum over the values of $\ell$ that are multiples of $q$ and belong to $\llbracket 1, n \rrbracket$.)

Let $$A = \frac{1}{8} \left(\begin{array}{llll} 9 & 1 & 3 & 3 \\ 1 & 9 & 3 & 3 \\ 3 & 3 & 9 & 1 \\ 3 & 3 & 1 & 9 \end{array}\right).$$ Construct an orthogonal and symplectic matrix $P$ such that $P^{\top} A P$ is diagonal.

Let $M \in \mathcal{A}_{2n}(\mathbb{R}) \cap \mathrm{Sp}_{2n}(\mathbb{R})$. Let $m$ be the linear map canonically associated with $M$. Show the equality $\mathrm{sp}_{\mathbb{R}}(M) = \emptyset$.

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 $k \in \{1, \ldots, n\}$, $d_{k+n} = 1/d_{k}$.

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.

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 $E$ 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.

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

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 $k = 4$.

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.

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 $u^N = \operatorname{Id}_E$.

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 $J_{n}$.

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.

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 $E$ be a $\mathbb{C}$-vector space of dimension $n \geq 1$. Let $u$ be a diagonalizable endomorphism of $E$. Show that $u$ is a permutation endomorphism if and only if there exist natural integers $c_1, \ldots, c_n$ such that, for all $k \in \mathbb{N}$,
$$\operatorname{Tr}\left(u^k\right) = \sum_{\substack{\ell=1 \\ \ell \mid k}}^{n} \ell c_\ell$$
(We sum over the values of $\ell$ dividing $k$ and belonging to $\llbracket 1, n \rrbracket$.)

We define the Redheffer matrix by $H_n = \left(h_{ij}\right)_{(i,j) \in \llbracket 1,n \rrbracket^2}$ where
$$h_{ij} = \begin{cases} 1 & \text{if } j = 1 \\ 1 & \text{if } i \text{ divides } j \text{ and } j \neq 1 \\ 0 & \text{otherwise.} \end{cases}$$
We also define the Mertens function $M$, by setting, for all $n \in \mathbb{N}^*, M(n) = \sum_{k=1}^{n} \mu(k)$ where $\mu$ is the Möbius function.
Let $A_n = \left(a_{ij}\right)_{(i,j) \in \llbracket 1,n \rrbracket^2}$ be the matrix with general term
$$a_{ij} = \begin{cases} \mu(j) & \text{if } i = 1 \\ 1 & \text{if } i = j \\ 0 & \text{otherwise} \end{cases}$$
and $C_n = A_n H_n$. By computing the coefficients of $C_n$, show that $\operatorname{det} H_n = M(n)$.
For the computation of the term with index $(i,j)$ of $C_n$, one may distinguish the case $i = j = 1$, the case $i = 1, j > 1$ and the case $i > 1, j > 1$.

Prove that a matrix $A \in \mathcal{M}_{n}(\mathbb{R})$ is orthodiagonalizable if and only if it is symmetric.

We set $A_1 = \left(\begin{array}{ccc} 3 & -2 & 4 \\ -2 & 6 & 2 \\ 4 & 2 & 3 \end{array}\right)$.
By observing the first and last column of $A_1$, determine an eigenvector of $A_1$ and the associated eigenvalue $\lambda_1$.

We set $A_1 = \left(\begin{array}{ccc} 3 & -2 & 4 \\ -2 & 6 & 2 \\ 4 & 2 & 3 \end{array}\right)$.
Determine the eigenspace of $A_1$ associated with the eigenvalue $\lambda_1$ and deduce the spectrum of $A_1$.

We denote by $G_0$ the subgroup of $G$ formed by elements $g$ such that $g(\mathcal{H})=\mathcal{H}$. For all $w\in V$ such that $B(w,w)>0$, we define the linear map $$s_w : v \mapsto v - 2\frac{B(v,w)}{B(w,w)}w.$$ Show that $s_w^2 = \mathrm{Id}_V$, and determine the eigenvalues and eigenspaces of $s_w$.