For all $n \in \mathbb{N}^*$, we define the matrix $H_n$ by: $$\forall (i,j) \in \llbracket 1; n \rrbracket^2, \quad (H_n)_{i,j} = \frac{1}{i+j-1}$$ We extend to $C^0([0;1], \mathbb{R})$ the inner product $\langle \cdot, \cdot \rangle$ by setting $$\forall f, g \in C^0([0;1], \mathbb{R}), \quad \langle f, g \rangle = \int_0^1 f(t) g(t) \, dt$$ and we denote by $\|\cdot\|$ the associated norm. For each $n \in \mathbb{N}$, $\Pi_n$ denotes the unique polynomial in $\mathbb{R}_n[X]$ minimizing $\|Q - f\|$ over $\mathbb{R}_n[X]$.
Show that the sequence $\left(\left\|\Pi_n - f\right\|\right)_{n \in \mathbb{N}}$ is decreasing and converges to 0.

For all $n \in \mathbb{N}^*$, we define the matrix $H_n$ by: $$\forall (i,j) \in \llbracket 1; n \rrbracket^2, \quad (H_n)_{i,j} = \frac{1}{i+j-1}$$ We extend to $C^0([0;1], \mathbb{R})$ the inner product $\langle \cdot, \cdot \rangle$ by setting $$\forall f, g \in C^0([0;1], \mathbb{R}), \quad \langle f, g \rangle = \int_0^1 f(t) g(t) \, dt$$
Show that $H_n$ is the matrix of the inner product $\langle \cdot, \cdot \rangle$, restricted to $\mathbb{R}_{n-1}[X]$, in the canonical basis of $\mathbb{R}_{n-1}[X]$.

For all $n \in \mathbb{N}^*$, we define the matrix $H_n$ by: $$\forall (i,j) \in \llbracket 1; n \rrbracket^2, \quad (H_n)_{i,j} = \frac{1}{i+j-1}$$ We extend to $C^0([0;1], \mathbb{R})$ the inner product $\langle \cdot, \cdot \rangle$ by setting $$\forall f, g \in C^0([0;1], \mathbb{R}), \quad \langle f, g \rangle = \int_0^1 f(t) g(t) \, dt$$ For each $n \in \mathbb{N}$, $\Pi_n$ denotes the unique polynomial in $\mathbb{R}_n[X]$ minimizing $\|Q - f\|$ over $\mathbb{R}_n[X]$.
Calculate the coefficients of $\Pi_n$ using the matrix $H_{n+1}^{-1}$ and the reals $\langle f, X^i \rangle$.

For all $n \in \mathbb{N}^*$, we define the matrix $H_n$ by: $$\forall (i,j) \in \llbracket 1; n \rrbracket^2, \quad (H_n)_{i,j} = \frac{1}{i+j-1}$$ We extend to $C^0([0;1], \mathbb{R})$ the inner product $\langle \cdot, \cdot \rangle$ by setting $$\forall f, g \in C^0([0;1], \mathbb{R}), \quad \langle f, g \rangle = \int_0^1 f(t) g(t) \, dt$$ For each $n \in \mathbb{N}$, $\Pi_n$ denotes the unique polynomial in $\mathbb{R}_n[X]$ minimizing $\|Q - f\|$ over $\mathbb{R}_n[X]$.
Determine explicitly $\Pi_2$ when $f$ is the function defined for all $t \in [0,1]$ by $f(t) = \frac{1}{1+t^2}$.

For $n \in \mathbb{N}^*$ and $(i,j) \in \llbracket 1, n \rrbracket^2$, we denote by $h_{i,j}^{(-1,n)}$ the coefficient at position $(i,j)$ of the matrix $H_n^{-1}$ and we denote by $s_n$ the sum of the coefficients of the matrix $H_n^{-1}$, that is: $$s_n = \sum_{1 \leqslant i,j \leqslant n} h_{i,j}^{(-1,n)}$$
Calculate $s_1$, $s_2$ and $s_3$. Conjecture in general the value of $s_n$ as a function of $n$.

For $n \in \mathbb{N}^*$ and $(i,j) \in \llbracket 1, n \rrbracket^2$, we denote by $h_{i,j}^{(-1,n)}$ the coefficient at position $(i,j)$ of the matrix $H_n^{-1}$ and we denote by $s_n$ the sum of the coefficients of the matrix $H_n^{-1}$, that is: $$s_n = \sum_{1 \leqslant i,j \leqslant n} h_{i,j}^{(-1,n)}$$
Let $n \in \mathbb{N}^*$.
a) Show that there exists a unique $n$-tuple of real numbers $\left(a_p^{(n)}\right)_{0 \leqslant p \leqslant n-1}$ satisfying the following system of $n$ linear equations in $n$ unknowns: $$\left\{\begin{array}{ccccccc} a_0^{(n)} + & \frac{a_1^{(n)}}{2} + \cdots + \frac{a_{n-1}^{(n)}}{n} = & 1 \\ \frac{a_0^{(n)}}{2} + \frac{a_1^{(n)}}{3} + \cdots + \frac{a_{n-1}^{(n)}}{n+1} = & 1 \\ \vdots & \vdots & & \vdots \\ \frac{a_0^{(n)}}{n} + \frac{a_1^{(n)}}{n+1} + \cdots + \frac{a_{n-1}^{(n)}}{2n-1} = & 1 \end{array}\right.$$
b) Show that $s_n = \sum_{p=0}^{n-1} a_p^{(n)}$.

For $n \in \mathbb{N}^*$ and $(i,j) \in \llbracket 1, n \rrbracket^2$, we denote by $h_{i,j}^{(-1,n)}$ the coefficient at position $(i,j)$ of the matrix $H_n^{-1}$ and we denote by $s_n$ the sum of the coefficients of the matrix $H_n^{-1}$, that is: $$s_n = \sum_{1 \leqslant i,j \leqslant n} h_{i,j}^{(-1,n)}$$ We define, for all $n \in \mathbb{N}^*$, the polynomial $S_n$ by: $S_n = a_0^{(n)} + a_1^{(n)} X + \cdots + a_{n-1}^{(n)} X^{n-1}$, where $\left(a_p^{(n)}\right)_{0 \leqslant p \leqslant n-1}$ is the unique solution of the system in IV.A.2.
Show that $$\forall Q = \alpha_0 + \alpha_1 X + \cdots + \alpha_{n-1} X^{n-1} \in \mathbb{R}_{n-1}[X], \quad \langle S_n, Q \rangle = \sum_{p=0}^{n-1} \alpha_p$$

For $n \in \mathbb{N}^*$ and $(i,j) \in \llbracket 1, n \rrbracket^2$, we denote by $h_{i,j}^{(-1,n)}$ the coefficient at position $(i,j)$ of the matrix $H_n^{-1}$. For $n \in \mathbb{N}$ and $k \in \llbracket 0; n \rrbracket$, we denote by $\binom{n}{k}$ the binomial coefficient $\binom{n}{k} = \frac{n!}{k!(n-k)!}$. The family $\left(K_p\right)_{p \in \mathbb{N}}$ is the orthonormal family defined in question II.E, and $K_n = \sqrt{2n+1}\,\Lambda_n$ where $\Lambda_n$ is a polynomial with integer coefficients.
Let $n \in \mathbb{N}^*$.
a) Calculate $h_{i,i}^{(-1,n)}$ for all $i \in \llbracket 1; n \rrbracket$; we will give in particular a very simple expression of $h_{1,1}^{(-1,n)}$ and $h_{n,n}^{(-1,n)}$ as a function of $n$.
b) Calculate $h_{i,j}^{(-1,n)}$ for all pairs $(i,j) \in \llbracket 1; n \rrbracket^2$; deduce that the coefficients of $H_n^{-1}$ are integers.
c) Show that $h_{i,j}^{(-1,n)}$ is divisible by 4 for all pairs $(i,j) \in \llbracket 2; n \rrbracket^2$.

Let $U_1, U_2, \cdots, U_n$, $n$ elements of $\mathbb{R}^p$ satisfying $\sum_{i=1}^n U_i = 0$. We define the matrix of squared mutual distances $M = \left(\|U_i - U_j\|^2\right)_{(i,j) \in \llbracket 1,n\rrbracket^2} \in \mathcal{S}_n(\mathbb{R})$. We denote $U$ the matrix of $\mathcal{M}_{p,n}(\mathbb{R})$ having as column vectors the elements $U_1, U_2, \cdots, U_n$.
Show that ${}^t UU = -\frac{1}{2}\Phi(M)$.

Let $U_1, U_2, \cdots, U_n$, $n$ elements of $\mathbb{R}^p$ satisfying $\sum_{i=1}^n U_i = 0$. We define the matrix of squared mutual distances $M = \left(\|U_i - U_j\|^2\right)_{(i,j) \in \llbracket 1,n\rrbracket^2} \in \mathcal{S}_n(\mathbb{R})$. It has been shown that ${}^t UU = -\frac{1}{2}\Phi(M)$.
Deduce, for every pair $(i,j) \in \llbracket 1,n\rrbracket^2$, an expression for the inner product $\langle U_i, U_j\rangle = {}^t U_i U_j$ as a function of $$\alpha_{ij} = -\frac{1}{n}\left(S(M)_i + S(M)_j\right) + \frac{1}{n^2}\sigma(M)$$ and of $m_{ij}$ (Torgerson relation).

We consider four points $U_1, U_2, U_3, U_4$ in $\mathbb{R}^3$ satisfying $U_1U_2 = U_2U_3 = U_3U_4 = U_4U_1 = 1$, $U_1U_3 = a$ and $U_2U_4 = b$. We use the notations of the previous parts with $n = 4$.
We set $M = \left(\|U_i - U_j\|^2\right)_{(i,j) \in \llbracket 1,4\rrbracket^2} \in \mathcal{S}_4(\mathbb{R})$.
Write the matrix $M$ then calculate $S(M)$ and $\sigma(M)$.

We consider four points $U_1, U_2, U_3, U_4$ in $\mathbb{R}^3$ satisfying $U_1U_2 = U_2U_3 = U_3U_4 = U_4U_1 = 1$, $U_1U_3 = a$ and $U_2U_4 = b$. We set $\Psi(M) = -\frac{1}{2}\Phi(M)$.
Show that the vectors $$\left(\begin{array}{r}1\\0\\-1\\0\end{array}\right), \left(\begin{array}{r}0\\1\\0\\-1\end{array}\right), \left(\begin{array}{r}-1\\1\\-1\\1\end{array}\right), \left(\begin{array}{l}1\\1\\1\\1\end{array}\right)$$ form a basis of eigenvectors of the matrix $\Psi(M)$ and determine the eigenvalues of the matrix $\Psi(M)$.

We consider four points $U_1, U_2, U_3, U_4$ in $\mathbb{R}^3$ satisfying $U_1U_2 = U_2U_3 = U_3U_4 = U_4U_1 = 1$, $U_1U_3 = a$ and $U_2U_4 = b$. We set $\Psi(M) = -\frac{1}{2}\Phi(M)$.
Determine the rank of $\Psi(M)$ according to the values taken by $a$ and $b$.

We consider four points $U_1, U_2, U_3, U_4$ in $\mathbb{R}^3$ satisfying $U_1U_2 = U_2U_3 = U_3U_4 = U_4U_1 = 1$, $U_1U_3 = a$ and $U_2U_4 = b$. We set $\Psi(M) = -\frac{1}{2}\Phi(M)$.
What equality do the reals $a$ and $b$ satisfy when the points $U_1, U_2, U_3$ and $U_4$ are coplanar?

We consider four points $U_1, U_2, U_3, U_4$ in $\mathbb{R}^3$ satisfying $U_1U_2 = U_2U_3 = U_3U_4 = U_4U_1 = 1$, $U_1U_3 = a$ and $U_2U_4 = b$. We set $\Psi(M) = -\frac{1}{2}\Phi(M)$.
Recover that the strictly positive reals $a$ and $b$ satisfy $a^2 + b^2 \leqslant 4$.

We consider four points $U_1, U_2, U_3, U_4$ in $\mathbb{R}^3$ satisfying $U_1U_2 = U_2U_3 = U_3U_4 = U_4U_1 = 1$, $U_1U_3 = a$ and $U_2U_4 = b$. We set $\Psi(M) = -\frac{1}{2}\Phi(M)$.
Conversely, if $a^2 + b^2 \leqslant 4$, give a family of points $U_1, U_2, U_3$ and $U_4$ satisfying the mutual distance constraints.

We consider a matrix $M = (m_{ij}) \in \mathcal{S}_n(\mathbb{R})$ such that for every $(i,j) \in \llbracket 1,n\rrbracket^2$, $m_{ij} \geqslant 0$ and $m_{ii} = 0$. We assume that $\Psi(M)$ has at least one strictly negative eigenvalue.
We seek to prove that there exists a unique symmetric matrix $T_0$ with non-negative eigenvalues that minimizes $\|\Psi(M) - T\|_{\mathcal{M}_n(\mathbb{R})}$ when $T$ ranges over $\mathcal{S}_n^+(\mathbb{R})$.
a) Show that $$\forall Q \in \mathcal{O}_n(\mathbb{R}), \forall A \in \mathcal{M}_n(\mathbb{R}), \quad \|{}^t QAQ\|_{\mathcal{M}_n(\mathbb{R})} = \|A\|_{\mathcal{M}_n(\mathbb{R})}$$
b) Justify the existence of a matrix $Q_0 \in \mathcal{O}_n(\mathbb{R})$ such that the matrix ${}^t Q_0 \Psi(M) Q_0$ is diagonal.
c) Show that a necessary condition for $\|\Psi(M) - T_0\|_{\mathcal{M}_n(\mathbb{R})}$ to minimize $\|\Psi(M) - T\|_{\mathcal{M}_n(\mathbb{R})}$ when $T$ ranges over $\mathcal{S}_n^+(\mathbb{R})$ is that the matrix ${}^t Q_0 T_0 Q_0$ is diagonal.
d) Prove the existence and uniqueness of the matrix $T_0$ sought.

We consider a matrix $M = (m_{ij}) \in \mathcal{S}_n(\mathbb{R})$ such that for every $(i,j) \in \llbracket 1,n\rrbracket^2$, $m_{ij} \geqslant 0$ and $m_{ii} = 0$. We assume that $\Psi(M)$ has at least one strictly negative eigenvalue. Let $T_0$ be the unique symmetric matrix with non-negative eigenvalues minimizing $\|\Psi(M) - T\|_{\mathcal{M}_n(\mathbb{R})}$ over $\mathcal{S}_n^+(\mathbb{R})$.
We assume in this question that $T_0$ is non-zero. We want to show that there exists a minimal integer $p \in \llbracket 1, n-1\rrbracket$ that we will specify such that we can determine vectors $U_1, U_2, \cdots, U_n$ elements of $\mathbb{R}^p$ satisfying the condition $\sum_{i=1}^n U_i = 0$ and for which the matrix $\widetilde{M} = \left(\|U_i - U_j\|^2\right)_{(i,j) \in \llbracket 1,n\rrbracket^2}$ satisfies the relation $\Psi(\widetilde{M}) = T_0$.
We use the notations of part II and denote $U = (U_1 | U_2 | \cdots | U_n)$.
a) Show that the integer $p$ satisfies $p \geqslant \operatorname{rg}(T_0)$ and that $\operatorname{rg}(T_0) \in \llbracket 1, n-1\rrbracket$.
b) Construct a matrix $U \in \mathcal{M}_{r,n}(\mathbb{R})$ such that ${}^t UU = T_0$ for $r = \operatorname{rg}(T_0)$.
Hint. Assuming that ${}^t Q_0 T_0 Q_0$ is of the form $\left(\begin{array}{ll}\Delta & \\ & 0_{n-r}\end{array}\right)$ with $\Delta \in \mathcal{M}_r(\mathbb{R})$, diagonal with non-zero values, we will seek $U$ in the form $U = \left((\Delta_1)(0)\right) \times Q_0 \in \mathcal{M}_{r,n}(\mathbb{R})$ with $\Delta_1 \in \mathcal{M}_r(\mathbb{R})$, diagonal.
c) Show that $\sum_{i=1}^n U_i = 0$ (we may study the vector $UZ$).
d) Deduce that $\Psi(\widetilde{M}) = T_0$ with $\widetilde{M} = \left(\|U_i - U_j\|^2\right)_{(i,j) \in \llbracket 1,n\rrbracket^2}$ and conclude.

We set $\xi_i^j = \begin{cases}0 & \text{if } i = j \\ 1 & \text{otherwise}\end{cases}$ and $N_k = (m_{ij} + k\xi_i^j)$ with $k$ a strictly positive real number.
Let $A \in \mathcal{M}_n(\mathbb{R})$. Show that the hyperplane $\mathcal{H}$ with normal vector $Z$ (and equation $x_1 + \cdots + x_n = 0$) is stable under the canonical endomorphism associated with the matrix $\Psi(A)$.

We set $\xi_i^j = \begin{cases}0 & \text{if } i = j \\ 1 & \text{otherwise}\end{cases}$ and $N_k = (m_{ij} + k\xi_i^j)$ with $k$ a strictly positive real number.
Express the matrix $N_k$ as a function of the matrices $M, J, I_n$ and the real $k$.

We set $\xi_i^j = \begin{cases}0 & \text{if } i = j \\ 1 & \text{otherwise}\end{cases}$ and $N_k = (m_{ij} + k\xi_i^j)$ with $k$ a strictly positive real number.
Show that there exists a minimal real $k_0$ that we will specify as a function of the eigenvalues of $\Psi(M)$, such that the matrix $\Psi(N_{k_0})$ has non-negative eigenvalues.

We set $D = (d_{ij})_{(i,j) \in \llbracket 1,n\rrbracket^2} = (\sqrt{m_{ij}})_{(i,j) \in \llbracket 1,n\rrbracket^2} \in \mathcal{M}_n(\mathbb{R})$ and $M_c = \left((d_{ij} + c\xi_i^j)^2\right)$ with $c > 0$, where $\xi_i^j = \begin{cases}0 & \text{if } i = j \\ 1 & \text{otherwise}\end{cases}$.
Show that, for all $X \in \mathbb{R}^n$, $${}^t X \Psi(M_c) X = {}^t X \Psi(M) X + 2c\, {}^t X \Psi(D) X + \frac{c^2}{2}\, {}^t X P X$$

We set $D = (d_{ij})_{(i,j) \in \llbracket 1,n\rrbracket^2} = (\sqrt{m_{ij}})_{(i,j) \in \llbracket 1,n\rrbracket^2} \in \mathcal{M}_n(\mathbb{R})$ and $M_c = \left((d_{ij} + c\xi_i^j)^2\right)$ with $c > 0$. The hyperplane $\mathcal{H}$ has normal vector $Z$ and equation $x_1 + \cdots + x_n = 0$.
Show that if $\lambda_{\min}$ and $\mu_{\min}$ denote the respective minimal eigenvalues of $\Psi(M)$ and $\Psi(D)$, then $$\forall X \in \mathcal{H}, \quad {}^t X \Psi(M) X \geqslant \lambda_{\min}\, {}^t XX \quad \text{and} \quad {}^t X \Psi(D) X \geqslant \mu_{\min}\, {}^t XX$$

We set $D = (d_{ij})_{(i,j) \in \llbracket 1,n\rrbracket^2} = (\sqrt{m_{ij}})_{(i,j) \in \llbracket 1,n\rrbracket^2} \in \mathcal{M}_n(\mathbb{R})$ and $M_c = \left((d_{ij} + c\xi_i^j)^2\right)$ with $c > 0$. Let $\lambda_{\min}$ and $\mu_{\min}$ denote the respective minimal eigenvalues of $\Psi(M)$ and $\Psi(D)$.
Deduce that for $c = \widetilde{c} = -2\mu_{\min} + \sqrt{4\mu_{\min}^2 - 2\lambda_{\min}} > 0$, $\Psi(M_c)$ has non-negative eigenvalues and that for all $c > \widetilde{c}$ and for all non-zero vector $X \in \mathcal{H}$, ${}^t X \Psi(M_c) X > 0$.

We set $D = (d_{ij})_{(i,j) \in \llbracket 1,n\rrbracket^2} = (\sqrt{m_{ij}})_{(i,j) \in \llbracket 1,n\rrbracket^2} \in \mathcal{M}_n(\mathbb{R})$ and $M_c = \left((d_{ij} + c\xi_i^j)^2\right)$ with $c > 0$. We seek the minimal constant $c^* > 0$ (if it exists) satisfying:

• $\Psi(M_{c^*})$ has non-negative eigenvalues,
• for all $c > c^*$ and for all non-zero vector $X \in \mathcal{H}$, ${}^t X \Psi(M_c) X > 0$.

We know that $c^*$ is bounded above by $\widetilde{c}$.
We consider $\mathcal{A} = \left\{X \in \mathcal{H} \mid \|X\| = 1 \text{ and } 4\left({}^t X \Psi(D) X\right)^2 - 2\, {}^t X \Psi(M) X \geqslant 0\right\}$ and we define the mapping $$\alpha: \begin{cases}\mathcal{A} \longrightarrow \mathbb{R} \\ X \longmapsto -2\, {}^t X \Psi(D) X + \sqrt{4\left({}^t X \Psi(D) X\right)^2 - 2\, {}^t X \Psi(M) X}\end{cases}$$
Show that there exists $X^* \in \mathcal{A}$ such that $\alpha(X^*) = \sup_{X \in \mathcal{A}} \alpha(X)$ and $\alpha(X^*) > 0$.