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 $D$.

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 $\mathcal{Z} := \{u \in \mathcal{W} : \bar{u} = 0\}$. Given $a \in E$ and $x \in E$, $(a \otimes x)(z) = (a \mid z) \cdot x$ for all $z \in E$.
Show that there exists a vector subspace $L$ of $E$ such that
$$\mathcal{Z} = \{a \otimes x \mid a \in L\} \quad \text{and} \quad \operatorname{dim} L = \operatorname{dim} \mathcal{Z},$$
and show that then $x \in L^{\perp}$.

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 $m_{ij}' = \begin{cases} g(j) & \text{if } j \text{ divides } i, \\ 0 & \text{otherwise.} \end{cases}$
Using the result $M = M' D^\top$, deduce that the determinant of $M$ equals
$$\operatorname{det} M = \prod_{k=1}^{n} g(k)$$

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 $\lambda$. We assume that $|x| < A|x|$. Show that there exists $\varepsilon > 0$ such that $A^2|x| - A|x| > \varepsilon A|x|$.

Show that the determinant of a symplectic matrix equals either 1 or $-1$.
Recall: A matrix $M \in \mathcal{M}_{2n}(\mathbb{R})$ is symplectic if and only if $M^{\top} J_{n} M = J_{n}$.

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 set: $$M = \left( \begin{array} { c c c c c } 0 & 0 & \cdots & 0 & a _ { 0 } \\ 1 & 0 & \ddots & 0 & a _ { 1 } \\ 0 & \ddots & \ddots & \vdots & \vdots \\ \vdots & \ddots & \ddots & 0 & a _ { d - 2 } \\ 0 & \cdots & 0 & 1 & a _ { d - 1 } \end{array} \right)$$
Compute the characteristic polynomial of $M$.

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 and $a_i \in \mathbb{Q}$. We set: $$M = \left( \begin{array} { c c c c c } 0 & 0 & \cdots & 0 & a _ { 0 } \\ 1 & 0 & \ddots & 0 & a _ { 1 } \\ 0 & \ddots & \ddots & \vdots & \vdots \\ \vdots & \ddots & \ddots & 0 & a _ { d - 2 } \\ 0 & \cdots & 0 & 1 & a _ { d - 1 } \end{array} \right)$$ Compute the characteristic polynomial of $M$.

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 $\mathcal{S}$, $\mathcal{S}_*$, $\mathcal{H}_0$, $(\mid)_{\mathcal{H}}$, $\gamma_{2\lambda}$, $\tau_x$ as defined previously. We denote $\mathcal{H}_0^\perp = \{h \in \mathcal{H} \mid \forall h_0 \in \mathcal{H}_0\ (h \mid h_0)_{\mathcal{H}} = 0\}$ the orthogonal subspace to $\mathcal{H}_0$ in $\mathcal{H}$.
(a) Show that $\mathcal{S}_* = \mathcal{S} \cap \mathcal{H}_0^\perp$.
(b) Show that $\mathcal{H}_0^\perp$ contains the vector subspace of $\mathcal{H}$ spanned by the functions $\tau_{x_i}(\gamma_{2\lambda})$ for $i \in \llbracket 1,p \rrbracket$.

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 $m_{ij}' = \begin{cases} g(j) & \text{if } j \text{ divides } i, \\ 0 & \text{otherwise.} \end{cases}$
Using the result $M = M' D^\top$, deduce that the determinant of $M$ equals
$$\operatorname{det} M = \prod_{k=1}^{n} g(k)$$

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 consider the sets $\overline{\mathcal{V}} := \{\bar{u} \mid u \in \mathcal{W}\}$ and $\mathcal{Z} := \{u \in \mathcal{W} : \bar{u} = 0\}$. Given $a \in E$ and $x \in E$, $(a \otimes x)(z) = (a \mid z) \cdot x$ for all $z \in E$. There exists a vector subspace $L$ of $E$ such that $\mathcal{Z} = \{a \otimes x \mid a \in L\}$ and $x \in L^{\perp}$.
By considering $u$ and $a \otimes x$ for $u \in \mathcal{V}$ and $a \in L$, deduce from Lemma A that $\mathcal{V} x \subset L^{\perp}$, and that more generally $u^{k}(x) \in L^{\perp}$ for every $k \in \mathbf{N}$ and every $u \in \mathcal{V}$.

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 $\lambda$. We assume that $|x| < A|x|$ and that there exists $\varepsilon > 0$ such that $A^2|x| - A|x| > \varepsilon A|x|$. We set $B = \frac{1}{1+\varepsilon} A$. Show that for all $k \geqslant 1, B^k A|x| \geqslant A|x|$.

Show that the inverse of a symplectic matrix is a symplectic matrix.

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 $B ( X , Y ) = X ^ { T } S Y$.
We set: $$M = \left( \begin{array} { c c c c c } 0 & 0 & \cdots & 0 & a _ { 0 } \\ 1 & 0 & \ddots & 0 & a _ { 1 } \\ 0 & \ddots & \ddots & \vdots & \vdots \\ \vdots & \ddots & \ddots & 0 & a _ { d - 2 } \\ 0 & \cdots & 0 & 1 & a _ { d - 1 } \end{array} \right)$$
There exist $P \in \mathrm { GL } _ { d } ( \mathbb { Q } )$ and $q _ { 1 } , \ldots , q _ { d } \in \mathbb { Q }$, $q _ { i } > 0$, such that $S = P ^ { T } \cdot \operatorname { Diag } \left( q _ { 1 } , \ldots , q _ { d } \right) \cdot P$.
17a. Verify that the matrix $S M$ is symmetric.
17b. Deduce that the matrix $R M R ^ { - 1 }$ is symmetric where $R = \operatorname { Diag } \left( \sqrt { q _ { 1 } } , \ldots , \sqrt { q _ { d } } \right) \cdot P$.

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 and $a_i \in \mathbb{Q}$. We consider the matrix $S$ of size $d \times d$ whose coefficient $( i , j )$ equals $t \left( z ^ { i + j } \right)$, and $P \in \mathrm{GL}_d(\mathbb{Q})$, $q_1, \ldots, q_d \in \mathbb{Q}$, $q_i > 0$ such that $S = P^T \cdot \operatorname{Diag}(q_1, \ldots, q_d) \cdot P$. We set: $$M = \left( \begin{array} { c c c c c } 0 & 0 & \cdots & 0 & a _ { 0 } \\ 1 & 0 & \ddots & 0 & a _ { 1 } \\ 0 & \ddots & \ddots & \vdots & \vdots \\ \vdots & \ddots & \ddots & 0 & a _ { d - 2 } \\ 0 & \cdots & 0 & 1 & a _ { d - 1 } \end{array} \right)$$
17a. Verify that the matrix $S M$ is symmetric.
17b. Deduce that the matrix $R M R ^ { - 1 }$ is symmetric where $R = \operatorname { Diag } \left( \sqrt { q _ { 1 } } , \ldots , \sqrt { q _ { d } } \right) \cdot P$.

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. Let $\alpha \in \mathbf{R}^p$ (resp. $a \in \mathbf{R}^p$) be the vector with coordinates $(\alpha_i)_{i \in \llbracket 1,p \rrbracket}$ (resp. $(a_i)_{i \in \llbracket 1,p \rrbracket}$) and $h_\alpha = \sum_{i=1}^p \alpha_i \tau_{x_i}(\gamma_{2\lambda})$. The matrix $K$ is defined by $K_{ij} = \exp\left(-\frac{|x_i - x_j|^2}{2\lambda}\right)$ (here in the case $d=1$).
(a) Show that $h_\alpha$ is an interpolant if and only if $K\alpha = a$ where $K$ is the matrix introduced in question (6) (here in the case $d = 1$).
(b) Show that $K$ 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
$$H := \operatorname{Vect}(x)^{\perp}, \quad \mathcal{V} x := \{v(x) \mid v \in \mathcal{V}\}$$
There exists a vector subspace $L$ of $E$ such that $\mathcal{Z} = \{a \otimes x \mid a \in L\}$, $x \in L^{\perp}$, and $\mathcal{V} x \subset L^{\perp}$.
Justify that $\lambda x \notin \mathcal{V} x$ for every $\lambda \in \mathbf{R}^{*}$, and deduce from the two previous questions that
$$\operatorname{dim} \mathcal{V} x + \operatorname{dim} L \leq n-1$$

Throughout part II, $A \in \mathcal{M}_n(\mathbb{R})$ is a strictly positive matrix satisfying $\rho(A) = 1$. We set $B = \frac{1}{1+\varepsilon} A$ for some $\varepsilon > 0$. Determine $\lim_{k \rightarrow +\infty} B^k$.

Show that the product of two symplectic matrices is a symplectic matrix. Is the set $\mathrm{Sp}_{2n}(\mathbb{R})$ a vector subspace of $\mathcal{M}_{2n}(\mathbb{R})$?

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 $P ( z ) = 0$.
Construct a symmetric matrix with rational coefficients for which $z$ is an eigenvalue.

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 and $a_i \in \mathbb{Q}$. We consider the matrix $S$ of size $d \times d$ whose coefficient $( i , j )$ equals $t \left( z ^ { i + j } \right)$, and $P \in \mathrm{GL}_d(\mathbb{Q})$, $q_1, \ldots, q_d \in \mathbb{Q}$, $q_i > 0$ such that $S = P^T \cdot \operatorname{Diag}(q_1, \ldots, q_d) \cdot P$. We set $R = \operatorname{Diag}\left(\sqrt{q_1}, \ldots, \sqrt{q_d}\right) \cdot P$ and $M$ the companion matrix of $Z(X)$.
Construct a symmetric matrix with rational coefficients for which $z$ is an eigenvalue.

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 $\alpha_*$, $h_\alpha$, $K$, $a$, $\mathcal{S}_*$, $J_*$, $(\mid)_{\mathcal{H}}$ as defined previously.
Deduce that there exists $\alpha_* \in \mathbf{R}^p$ such that $\mathcal{S}_* = \{h_{\alpha_*}\}$ and calculate the value of $J_*$ in terms of $K$ and $a$.

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
$$H := \operatorname{Vect}(x)^{\perp}, \quad \mathcal{W} := \{v \in \mathcal{V} : v(x) = 0\}$$
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 set $\overline{\mathcal{V}} := \{\bar{u} \mid u \in \mathcal{W}\}$.
Let $u \in \mathcal{W}$. Show that $(\bar{u})^{k}(z) = \pi(u^{k}(z))$ for every $k \in \mathbf{N}$ and every $z \in H$. Deduce that $\overline{\mathcal{V}}$ is a nilpotent vector subspace of $\mathcal{L}(H)$.

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 $\lambda$. Conclude (that 1 is an eigenvalue of $A$).

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.

We fix a real vector space $E$ of dimension $n \geq 2$, 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 $H := \operatorname{Vect}(x)^{\perp}$, $\mathcal{V} x := \{v(x) \mid v \in \mathcal{V}\}$, $\mathcal{W} := \{v \in \mathcal{V} : v(x) = 0\}$, $\overline{\mathcal{V}} := \{\bar{u} \mid u \in \mathcal{W}\}$, $\mathcal{Z} := \{u \in \mathcal{W} : \bar{u} = 0\}$, and $L$ the vector subspace such that $\mathcal{Z} = \{a \otimes x \mid a \in L\}$.
We have $\operatorname{dim} \mathcal{V} = \operatorname{dim}(\mathcal{V} x) + \operatorname{dim} \mathcal{Z} + \operatorname{dim} \overline{\mathcal{V}}$, $\operatorname{dim} \mathcal{V} x + \operatorname{dim} L \leq n-1$, $\overline{\mathcal{V}}$ is a nilpotent subspace of $\mathcal{L}(H)$ with $\dim H = n-1$, and by induction hypothesis $\operatorname{dim} \overline{\mathcal{V}} \leq \frac{(n-1)(n-2)}{2}$.
Prove that
$$\operatorname{dim} \mathcal{V} \leq \frac{n(n-1)}{2}$$