Throughout the rest of the problem, $K = J_{n}$. For all $X = \left(\begin{array}{c} x_{1} \\ x_{2} \\ \vdots \\ x_{2n} \end{array}\right) \in \mathcal{M}_{2n,1}(\mathbb{R})$ and for all $Y = \left(\begin{array}{c} y_{1} \\ y_{2} \\ \vdots \\ y_{2n} \end{array}\right) \in \mathcal{M}_{2n,1}(\mathbb{R})$, where $\varphi(X,Y) = X^{\top} J_{n} Y$, show the equality $$\varphi(X,Y) = \sum_{k=1}^{n} \left(x_{k} y_{k+n} - x_{k+n} y_{k}\right).$$

We consider an $\mathbf{R}$-vector space $E$ of dimension $n > 0$. Let $\mathcal{V}$ be a nilpotent vector subspace of $\mathcal{L}(E)$ containing a non-zero element, with generic nilindex $p := \max_{u \in \mathcal{V}} \nu(u)$. In questions 8 to 11, we are given two arbitrary elements $u$ and $v$ of $\mathcal{V}$.
Show that $\sum_{i=0}^{p-1} u^{i} v u^{p-1-i} = 0$.

With $K = J_{n}$ and $\varphi(X,Y) = X^{\top} J_{n} Y$: Show that for all $(i,j) \in \{1,\ldots,2n\}^{2}$, $$\varphi(e_{i}, e_{j}) = \delta_{i+n,j} - \delta_{i,j+n}$$ (one may start with the case where $(i,j) \in \{1,\ldots,n\}^{2}$ then generalize).

Throughout part II, $A \in \mathcal{M}_n(\mathbb{R})$ is a strictly positive matrix. Show that, for all $x \in \mathbb{R}^n$, $$\left\{\begin{array}{l} x \geqslant 0 \Longrightarrow Ax \geqslant 0 \\ x \geqslant 0 \text{ and } x \neq 0 \Longrightarrow Ax > 0. \end{array}\right.$$

With $K = J_{n}$ and $\varphi(X,Y) = X^{\top} J_{n} Y$: Show that for all $X \in \mathcal{M}_{2n,1}(\mathbb{R})$, $J_{n} X \in X^{\perp}$ and compute $\varphi(J_{n} X, X)$.

Throughout part II, $A \in \mathcal{M}_n(\mathbb{R})$ is a strictly positive matrix. Show that $\forall k \in \mathbb{N}^*, A^k > 0$.

With $K = J_{n}$ and $\varphi(X,Y) = X^{\top} J_{n} Y$: If $Y \in \mathcal{M}_{2n,1}(\mathbb{R})$, we denote by $Y^{J_{n}}$ the set of vectors $Z$ of $\mathcal{M}_{2n,1}(\mathbb{R})$ such that $\varphi(Y,Z) = 0$. Show that $X^{J_{n}} = (J_{n} X)^{\perp}$.

We consider an $\mathbf{R}$-vector space $E$ of dimension $n > 0$. Let $\mathcal{V}$ be a nilpotent vector subspace of $\mathcal{L}(E)$ containing a non-zero element, with generic nilindex $p := \max_{u \in \mathcal{V}} \nu(u)$. We introduce the subset $\mathcal{V}^{\bullet}$ of $E$ formed by vectors belonging to at least one of the sets $\operatorname{Im} u^{p-1}$ for $u$ in $\mathcal{V}$, the vector subspace $K(\mathcal{V}) := \operatorname{Vect}(\mathcal{V}^{\bullet})$, and given $x \in E$, $\mathcal{V} x := \{v(x) \mid v \in \mathcal{V}\}$.
Lemma B states: Let $x$ be in $\mathcal{V}^{\bullet} \backslash \{0\}$. If $K(\mathcal{V}) \subset \operatorname{Vect}(x) + \mathcal{V} x$, then $v(x) = 0$ for every $v$ in $\mathcal{V}$.
Let $x \in \mathcal{V}^{\bullet} \backslash \{0\}$ such that $K(\mathcal{V}) \subset \operatorname{Vect}(x) + \mathcal{V} x$. We choose $u \in \mathcal{V}$ such that $x \in \operatorname{Im} u^{p-1}$.
Given $y \in K(\mathcal{V})$, show that for every $k \in \mathbf{N}$ there exist $y_{k} \in K(\mathcal{V})$ and $\lambda_{k} \in \mathbf{R}$ such that $y = \lambda_{k} x + u^{k}(y_{k})$. Deduce that $K(\mathcal{V}) \subset \operatorname{Vect}(x)$ and then that $v(x) = 0$ for every $v \in \mathcal{V}$.

Throughout part II, $A \in \mathcal{M}_n(\mathbb{R})$ is a strictly positive matrix. Deduce that $\rho(A) > 0$ then show that $\rho\left(\frac{A}{\rho(A)}\right) = 1$.

With $K = J_{n}$ and $\varphi(X,Y) = X^{\top} J_{n} Y$: Let $P$ be a symplectic and orthogonal matrix whose columns are denoted $X_{1}, \ldots, X_{2n}$. Show that, for all $(i,j) \in \{1,\ldots,2n\}^{2}$, $$\left\{\begin{array}{l} \|X_{i}\| = 1 \\ i \neq j \Longrightarrow X_{i} \perp X_{j} \\ \varphi(X_{i}, X_{j}) = \delta_{i+n,j} - \delta_{i,j+n} \end{array}\right.$$

We denote by $\mathscr { R }$ the set of totally real numbers and we admit that there exists a function $t : \mathscr { R } \rightarrow \mathbb { Q }$ satisfying the following two properties: (i) for $x , y \in \mathscr { R }$ and $\lambda , \mu \in \mathbb { Q }$, we have $t ( \lambda x + \mu y ) = \lambda t ( x ) + \mu t ( y )$ (ii) for $x$ totally positive, we have $t ( x ) \geqslant 0$ and the equality is strict if $x \neq 0$.
We consider a non-zero totally real number $z$. By definition, there exists a monic polynomial $Z ( X ) \in \mathbb { Q } [ X ]$ that annihilates $z$. We write $Z ( X )$ in the form: $$Z ( X ) = X ^ { d } - \left( a _ { d - 1 } X ^ { d - 1 } + \cdots + a _ { 1 } X + a _ { 0 } \right)$$ with $d \in \mathbb{N} ^ { * }$ and $a _ { i } \in \mathbb { Q }$ for all $i \in \{ 0 , \ldots , d - 1 \}$. We further assume that $Z ( X )$ is chosen so that $d$ is minimal among the degrees of monic polynomials $P ( X ) \in \mathbb{Q} [ X ]$ such that $P ( z ) = 0$. We consider the matrix $S$ of size $d \times d$ whose coefficient $(i, j)$, $1 \leqslant i , j \leqslant d$, equals $t ( z ^ { i + j } )$. For $X , Y \in \mathbb { R } ^ { d }$, we set $B ( X , Y ) = X ^ { T } S Y$.
13a. Show that $B ( X , X ) > 0$ for $X \in \mathbb { Q } ^ { d }$, $X \neq 0$.
13b. Deduce that the matrix $S$ is invertible.

We denote by $\mathscr { R }$ the set of totally real numbers and we admit that there exists a function $t : \mathscr { R } \rightarrow \mathbb { Q }$ satisfying the following two properties: (i) for $x , y \in \mathscr { R }$ and $\lambda , \mu \in \mathbb { Q }$, we have $t ( \lambda x + \mu y ) = \lambda t ( x ) + \mu t ( y )$ (ii) for $x$ totally positive, we have $t ( x ) \geqslant 0$ and the equality is strict if $x \neq 0$.
We consider a non-zero totally real number $z$. By definition, there exists a monic polynomial $Z ( X ) \in \mathbb { Q } [ X ]$ that annihilates $z$. We write $Z ( X )$ in the form: $$Z ( X ) = X ^ { d } - \left( a _ { d - 1 } X ^ { d - 1 } + \cdots + a _ { 1 } X + a _ { 0 } \right)$$ with $d \in \mathbb { N } ^ { \star }$ and $a _ { i } \in \mathbb { Q }$ for all $i \in \{ 0 , \ldots , d - 1 \}$. We further assume that $Z ( X )$ is chosen so that $d$ is minimal among the degrees of monic polynomials $P ( X ) \in \mathbb { Q } [ X ]$ such that $P ( z ) = 0$. We consider the matrix $S$ of size $d \times d$ whose coefficient $( i , j ) , 1 \leqslant i , j \leqslant d$, equals $t \left( z ^ { i + j } \right)$. For $X , Y \in \mathbb { R } ^ { d }$, we set $B ( X , Y ) = X ^ { T } S Y$.
13a. Show that $B ( X , X ) > 0$ for $X \in \mathbb { Q } ^ { d } , X \neq 0$.
13b. Deduce that the matrix $S$ is invertible.

We fix a real vector space $E$ of dimension $n$, as well as a nilpotent vector subspace $\mathcal{V}$ of $\mathcal{L}(E)$, equipped with an inner product $(-\mid-)$. We consider an arbitrary vector $x$ of $E \backslash \{0\}$, and set
$$H := \operatorname{Vect}(x)^{\perp}, \quad \mathcal{V} x := \{v(x) \mid v \in \mathcal{V}\} \text{ and } \mathcal{W} := \{v \in \mathcal{V} : v(x) = 0\}$$
We denote by $\pi$ the orthogonal projection of $E$ onto $H$. For $u \in \mathcal{W}$, we denote by $\bar{u}$ the endomorphism of $H$ defined by $\forall z \in H, \bar{u}(z) = \pi(u(z))$. We consider the sets $\overline{\mathcal{V}} := \{\bar{u} \mid u \in \mathcal{W}\}$ and $\mathcal{Z} := \{u \in \mathcal{W} : \bar{u} = 0\}$.
Show that $\mathcal{V} x$, $\mathcal{W}$, $\overline{\mathcal{V}}$ and $\mathcal{Z}$ are vector subspaces of $E$, $\mathcal{V}$, $\mathcal{L}(H)$ and $\mathcal{V}$ respectively.

Throughout part II, $A \in \mathcal{M}_n(\mathbb{R})$ is a strictly positive matrix. Assume $A$ is diagonalizable over $\mathbb{C}$. Show that, if $\rho(A) < 1$ then $\lim_{k \rightarrow +\infty} A^k = 0$.

With $K = J_{n}$, $\varphi(X,Y) = X^{\top} J_{n} Y$, and $P$ a symplectic and orthogonal matrix with columns $X_{1}, \ldots, X_{2n}$ satisfying the properties of Q13: Show that, for all $i \in \{1,\ldots,n\}$, $X_{i}^{J_{n}} = X_{i+n}^{\perp}$.

We denote by $\mathscr { R }$ the set of totally real numbers and we admit that there exists a function $t : \mathscr { R } \rightarrow \mathbb { Q }$ satisfying the following two properties: (i) for $x , y \in \mathscr { R }$ and $\lambda , \mu \in \mathbb { Q }$, we have $t ( \lambda x + \mu y ) = \lambda t ( x ) + \mu t ( y )$ (ii) for $x$ totally positive, we have $t ( x ) \geqslant 0$ and the equality is strict if $x \neq 0$.
We consider a non-zero totally real number $z$. By definition, there exists a monic polynomial $Z ( X ) \in \mathbb { Q } [ X ]$ that annihilates $z$. We write $Z ( X )$ in the form: $$Z ( X ) = X ^ { d } - \left( a _ { d - 1 } X ^ { d - 1 } + \cdots + a _ { 1 } X + a _ { 0 } \right)$$ with $d \in \mathbb{N} ^ { * }$ and $a _ { i } \in \mathbb { Q }$ for all $i \in \{ 0 , \ldots , d - 1 \}$. We further assume that $Z ( X )$ is chosen so that $d$ is minimal among the degrees of monic polynomials $P ( X ) \in \mathbb{Q} [ X ]$ such that $P ( z ) = 0$. We consider the matrix $S$ of size $d \times d$ whose coefficient $(i, j)$, $1 \leqslant i , j \leqslant d$, equals $t ( z ^ { i + j } )$. For $X , Y \in \mathbb { R } ^ { d }$, we set $B ( X , Y ) = X ^ { T } S Y$.
Show that $B$ is an inner product on $\mathbb { R } ^ { d }$.

We denote by $\mathscr { R }$ the set of totally real numbers and we admit that there exists a function $t : \mathscr { R } \rightarrow \mathbb { Q }$ satisfying the following two properties: (i) for $x , y \in \mathscr { R }$ and $\lambda , \mu \in \mathbb { Q }$, we have $t ( \lambda x + \mu y ) = \lambda t ( x ) + \mu t ( y )$ (ii) for $x$ totally positive, we have $t ( x ) \geqslant 0$ and the equality is strict if $x \neq 0$.
We consider a non-zero totally real number $z$. We write $Z ( X ) = X ^ { d } - \left( a _ { d - 1 } X ^ { d - 1 } + \cdots + a _ { 1 } X + a _ { 0 } \right)$ with $d$ minimal. We consider the matrix $S$ of size $d \times d$ whose coefficient $( i , j )$ equals $t \left( z ^ { i + j } \right)$. For $X , Y \in \mathbb { R } ^ { d }$, we set $B ( X , Y ) = X ^ { T } S Y$.
Show that $B$ is an inner product on $\mathbb { R } ^ { d }$.

We fix two $p$-tuples $(x_i)_{i \in \llbracket 1,p \rrbracket}$ and $(a_i)_{i \in \llbracket 1,p \rrbracket}$ of real numbers. We assume that the $x_i$ are pairwise distinct. We denote $\mathcal{S} = \{h \in \mathcal{H} \mid h(x_i) = a_i\}$ the set of $h \in \mathcal{H}$ that equal $a_i$ at $x_i$ for all $i \in \llbracket 1,p \rrbracket$. We denote $J : \mathcal{H} \rightarrow \mathbf{R}$ defined by $J(h) = \frac{1}{2}\|h\|_{\mathcal{H}}^2$ and $J_* = \inf\{J(h) \mid h \in \mathcal{S}\}$. We denote $\mathcal{S}_* = \{h \in \mathcal{S} \mid J(h) = J_*\}$.
Show that $\mathcal{S}_*$ has at most one element.

We fix a real vector space $E$ of dimension $n$, as well as a nilpotent vector subspace $\mathcal{V}$ of $\mathcal{L}(E)$, equipped with an inner product $(-\mid-)$. We consider an arbitrary vector $x$ of $E \backslash \{0\}$, and set
$$H := \operatorname{Vect}(x)^{\perp}, \quad \mathcal{V} x := \{v(x) \mid v \in \mathcal{V}\} \text{ and } \mathcal{W} := \{v \in \mathcal{V} : v(x) = 0\}$$
We denote by $\pi$ the orthogonal projection of $E$ onto $H$. For $u \in \mathcal{W}$, we denote by $\bar{u}$ the endomorphism of $H$ defined by $\forall z \in H, \bar{u}(z) = \pi(u(z))$. We consider the sets $\overline{\mathcal{V}} := \{\bar{u} \mid u \in \mathcal{W}\}$ and $\mathcal{Z} := \{u \in \mathcal{W} : \bar{u} = 0\}$.
Show that
$$\operatorname{dim} \mathcal{V} = \operatorname{dim}(\mathcal{V} x) + \operatorname{dim} \mathcal{Z} + \operatorname{dim} \overline{\mathcal{V}}.$$

Let $f$ be an arithmetic function, $n \in \mathbb{N}^*$ and $g = f * \mu$. We denote $M = \left(m_{ij}\right)$ the matrix of $\mathcal{M}_n(\mathbb{C})$ with general term $m_{ij} = f(i \wedge j)$. We also define the divisor matrix $D = \left(d_{ij}\right)$ by:
$$d_{ij} = \begin{cases} 1 & \text{if } j \text{ divides } i \\ 0 & \text{otherwise} \end{cases}$$
Let $M'$ be the matrix with general term $m_{ij}' = \begin{cases} g(j) & \text{if } j \text{ divides } i, \\ 0 & \text{otherwise.} \end{cases}$
Show that $M = M' D^\top$, where $D^\top$ is the transpose of $D$.

Throughout part II, $A \in \mathcal{M}_n(\mathbb{R})$ is a strictly positive matrix satisfying $\rho(A) = 1$. We consider an eigenvalue $\lambda \in \mathbb{C}$ of $A$ with modulus 1 and $x$ an eigenvector associated with $\lambda$. Show that $|x| \leqslant A|x|$.

With $K = J_{n}$, $\varphi(X,Y) = X^{\top} J_{n} Y$, and $P$ a symplectic and orthogonal matrix with columns $X_{1}, \ldots, X_{2n}$ satisfying the properties of Q13: Show that, for all $i \in \{1,\ldots,n\}$, $X_{i+n} = -J_{n} X_{i}$.

We denote by $\mathscr { R }$ the set of totally real numbers and we admit that there exists a function $t : \mathscr { R } \rightarrow \mathbb { Q }$ satisfying the following two properties: (i) for $x , y \in \mathscr { R }$ and $\lambda , \mu \in \mathbb { Q }$, we have $t ( \lambda x + \mu y ) = \lambda t ( x ) + \mu t ( y )$ (ii) for $x$ totally positive, we have $t ( x ) \geqslant 0$ and the equality is strict if $x \neq 0$.
We consider a non-zero totally real number $z$. By definition, there exists a monic polynomial $Z ( X ) \in \mathbb { Q } [ X ]$ that annihilates $z$. We write $Z ( X )$ in the form: $$Z ( X ) = X ^ { d } - \left( a _ { d - 1 } X ^ { d - 1 } + \cdots + a _ { 1 } X + a _ { 0 } \right)$$ with $d \in \mathbb{N} ^ { * }$ and $a _ { i } \in \mathbb { Q }$ for all $i \in \{ 0 , \ldots , d - 1 \}$. We further assume that $Z ( X )$ is chosen so that $d$ is minimal among the degrees of monic polynomials $P ( X ) \in \mathbb{Q} [ X ]$ such that $P ( z ) = 0$. We consider the matrix $S$ of size $d \times d$ whose coefficient $(i, j)$, $1 \leqslant i , j \leqslant d$, equals $t ( z ^ { i + j } )$. For $X , Y \in \mathbb { R } ^ { d }$, we set $B ( X , Y ) = X ^ { T } S Y$.
15a. Show that there exists a basis $(e _ { 1 } , \ldots , e _ { d })$ of $\mathbb { R } ^ { d }$ with $e _ { i } \in \mathbb { Q } ^ { d }$ for all $i$ and $B ( e _ { i } , e _ { j } ) = 0$ for $i \neq j$.
15b. Deduce that there exist $P \in \mathrm { GL } _ { d } ( \mathbb { Q } )$ and $q _ { 1 } , \ldots , q _ { d } \in \mathbb { Q }$, $q _ { i } > 0$, such that: $$S = P ^ { T } \cdot \operatorname { Diag } \left( q _ { 1 } , \ldots , q _ { d } \right) \cdot P$$

We denote by $\mathscr { R }$ the set of totally real numbers and we admit that there exists a function $t : \mathscr { R } \rightarrow \mathbb { Q }$ satisfying the following two properties: (i) for $x , y \in \mathscr { R }$ and $\lambda , \mu \in \mathbb { Q }$, we have $t ( \lambda x + \mu y ) = \lambda t ( x ) + \mu t ( y )$ (ii) for $x$ totally positive, we have $t ( x ) \geqslant 0$ and the equality is strict if $x \neq 0$.
We consider a non-zero totally real number $z$. We write $Z ( X ) = X ^ { d } - \left( a _ { d - 1 } X ^ { d - 1 } + \cdots + a _ { 1 } X + a _ { 0 } \right)$ with $d$ minimal. We consider the matrix $S$ of size $d \times d$ whose coefficient $( i , j )$ equals $t \left( z ^ { i + j } \right)$. For $X , Y \in \mathbb { R } ^ { d }$, we set $B ( X , Y ) = X ^ { T } S Y$.
15a. Show that there exists a basis $\left( e _ { 1 } , \ldots , e _ { d } \right)$ of $\mathbb { R } ^ { d }$ with $e _ { i } \in \mathbb { Q } ^ { d }$ for all $i$ and $B \left( e _ { i } , e _ { j } \right) = 0$ for $i \neq j$.
15b. Deduce that there exist $P \in \mathrm { GL } _ { d } ( \mathbb { Q } )$ and $q _ { 1 } , \ldots , q _ { d } \in \mathbb { Q } , q _ { i } > 0$, such that: $$S = P ^ { T } \cdot \operatorname { Diag } \left( q _ { 1 } , \ldots , q _ { d } \right) \cdot P$$

We fix two $p$-tuples $(x_i)_{i \in \llbracket 1,p \rrbracket}$ and $(a_i)_{i \in \llbracket 1,p \rrbracket}$ of real numbers with the $x_i$ pairwise distinct. We use the notation $\mathcal{S}$, $J$, $J_*$, $\mathcal{S}_*$, $(\mid)_{\mathcal{H}}$ as defined previously.
Let $\mathcal{H}_0 = \{h \in \mathcal{H} \mid h(x_i) = 0\ \forall i \in \llbracket 1,p \rrbracket\}$ and $\tilde{h} \in \mathcal{S}_*$ (we assume here $\mathcal{S}_*$ non-empty).
Show that $\left(\tilde{h} \mid h_0\right)_{\mathcal{H}} = 0$ for all $h_0 \in \mathcal{H}_0$.