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 $s_n = \sum_{p=0}^{n-1} a_p^{(n)}$.

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 $D = (d_{ij})_{(i,j) \in \llbracket 1,n\rrbracket^2} = (\sqrt{m_{ij}})_{(i,j) \in \llbracket 1,n\rrbracket^2} \in \mathcal{M}_n(\mathbb{R})$ 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 $X \in \mathbb{R}^n$, $${}^t X \Psi(M_c) X = {}^t X \Psi(M) X + 2c\, {}^t X \Psi(D) X + \frac{c^2}{2}\, {}^t X P X$$

Prove that the real symmetric matrix $A$ is invertible. (One may consider the kernel of the map $x \mapsto A x$).

Prove that $\forall x , y \in \mathbb { R } ^ { n } , \left\langle A ^ { - 1 } x ; y \right\rangle = \left\langle x ; A ^ { - 1 } y \right\rangle$. Deduce that the matrix $A ^ { - 1 }$ is symmetric.

For $x , y \in \mathbb { R } ^ { n }$, we set: $( x ; y ) _ { A } = \langle A x ; y \rangle$. We denote by $E$ the endomorphism of the vector space $\mathbb { R } ^ { n }$ defined by $\forall x \in \mathbb { R } ^ { n } , E ( x ) = A ^ { - 1 } K x$.
Prove that $( ; ) _ { A }$ defines an inner product on $\mathbb { R } ^ { n }$. Then show that $$\forall x , y \in \mathbb { R } ^ { n } , ( E ( x ) ; y ) _ { A } = ( x ; E ( y ) ) _ { A } .$$

$\mathcal{P}$ denotes the vector space of polynomial functions with complex coefficients. We denote for every pair $(P,Q) \in \mathcal{P}^2$, $$\langle P, Q \rangle = \int_0^{+\infty} e^{-t}\bar{P}(t)Q(t)\,dt.$$
Verify that $\langle \cdot, \cdot \rangle$ defines an inner product on $\mathcal{P}$.

$\mathcal{P}$ denotes the vector space of polynomial functions with complex coefficients, with inner product $\langle P, Q \rangle = \displaystyle\int_0^{+\infty} e^{-t}\bar{P}(t)Q(t)\,dt$.
We denote by $D$ the differentiation endomorphism and $U$ the endomorphism of $\mathcal{P}$ defined by $$U(P)(t) = e^t D\left(te^{-t}P^{\prime}(t)\right).$$
Verify that $U$ is an endomorphism of $\mathcal{P}$.

Let $x$ be a linear recurrent sequence of minimal order $p \geqslant 1$ and minimal polynomial $B$. For any integer $n$ in $\mathbb{N}^*$, we denote by $H_n(x)$ the matrix in $\mathcal{M}_n(\mathbb{K})$ defined by $\forall (i,j) \in \{1,\ldots,n\}^2, [H_n(x)]_{i,j} = x_{i+j-2}$.
Show that the family $(\sigma^k(x))_{0 \leqslant k \leqslant p-1}$ is a basis of $\mathcal{R}_B(\mathbb{K})$.
Deduce from this, for any $n$ in $\mathbb{N}^*$, the rank of the family $(\sigma^k(x))_{0 \leqslant k \leqslant n-1}$.

Let $x$ be a linear recurrent sequence of minimal order $p \geqslant 1$ and minimal polynomial $B$. For any integer $n$ in $\mathbb{N}^*$, we denote by $H_n(x)$ the matrix in $\mathcal{M}_n(\mathbb{K})$ defined by $\forall (i,j) \in \{1,\ldots,n\}^2, [H_n(x)]_{i,j} = x_{i+j-2}$.
Show that if $n \geqslant p$, the map $\varphi_n : \left\{ \begin{array}{l} \mathcal{R}_B(\mathbb{K}) \rightarrow \mathbb{K}^n \\ v \mapsto (v_0, \ldots, v_{n-1}) \end{array} \right.$ is injective.
Deduce from this that if $n \geqslant p$, then $\operatorname{rang}(H_n(x)) = p$.
Remark: it is clear that this result remains true if $p = 0$ (since the sequence $x$ and the matrices $H_n(x)$ are zero).

Let $x$ be a non-zero linear recurrent sequence, of order $m \geqslant 1$. Let $p = \operatorname{rang}(H_m(x))$. For any integer $n$ in $\mathbb{N}^*$, we denote by $H_n(x)$ the matrix in $\mathcal{M}_n(\mathbb{K})$ defined by $\forall (i,j) \in \{1,\ldots,n\}^2, [H_n(x)]_{i,j} = x_{i+j-2}$.
Show that $x$ is of minimal order $p$ and that the kernel of $H_{p+1}(x)$ is a one-dimensional vector space whose a direction vector can be written $(b_0, \ldots, b_{p-1}, 1)$, where $b_0, \ldots, b_{p-1}$ are in $\mathbb{K}$.

Throughout this part, $n$ denotes an integer greater than or equal to 3. We say that a matrix $M = (m_{i,j})_{1 \leqslant i,j \leqslant n}$ in $\mathcal{M}_n(\mathbb{R})$ is a Hankel matrix if there exists $a = (a_0, \ldots, a_{2n-2}) \in \mathbb{R}^{2n-1}$ such that for all $i$ and $j$ in $\{1, \ldots, n\}$, $m_{i,j} = a_{i+j-2}$. Such a matrix is denoted $M = H(a)$.
Show that if $M$ is a Hankel matrix of size $n$ then it admits $n$ real eigenvalues $\lambda_1, \ldots, \lambda_n$ (each repeated as many times as its multiplicity) which can be ordered in decreasing order $\lambda_1 \geqslant \lambda_2 \geqslant \ldots \geqslant \lambda_n$.

Throughout this part, $n$ denotes an integer greater than or equal to 3. We say that a matrix $M = (m_{i,j})_{1 \leqslant i,j \leqslant n}$ in $\mathcal{M}_n(\mathbb{R})$ is a Hankel matrix if there exists $a = (a_0, \ldots, a_{2n-2}) \in \mathbb{R}^{2n-1}$ such that for all $i$ and $j$ in $\{1, \ldots, n\}$, $m_{i,j} = a_{i+j-2}$. Such a matrix is denoted $M = H(a)$.
Show that if $\lambda \in \mathbb{R}^*$ then the $n$-tuple $(\lambda, \ldots, \lambda)$ is not the ordered $n$-tuple of eigenvalues of a Hankel matrix of size $n$.

Throughout this part, $n$ denotes an integer greater than or equal to 3, and $p = [(n+1)/2]$ is the integer part of $(n+1)/2$. Let $a = (a_0, \ldots, a_{2n-2})$ be an element of $\mathbb{R}^{2n-1}$ and $M = H(a)$. We denote $\operatorname{Spo}(M) = (\lambda_1, \ldots, \lambda_n)$.
We define two vectors $v = (v_1, \ldots, v_n)$ and $w = (w_1, \ldots, w_n)$ of $\mathbb{R}^n$ by $$\begin{cases} v_i = \sqrt{2i-1}\, a_{2(i-1)} \text{ and } w_i = \dfrac{1}{\sqrt{2i-1}} & \text{if } i \in \{1, \ldots, p\} \\ v_i = \sqrt{2n-2i+1}\, a_{2(i-1)} \text{ and } w_i = \dfrac{1}{\sqrt{2n-2i+1}} & \text{if } i \in \{p+1, \ldots, n\} \end{cases}$$ Finally, we set $K_n = n - \|w\|^2$.
Show that $$\sum_{i=1}^{n} \lambda_i = \sum_{k=0}^{n-1} a_{2k} \quad \text{and} \quad \sum_{i=1}^{n} \lambda_i^2 = \sum_{k=0}^{n-1} (k+1) a_k^2 + \sum_{k=n}^{2n-2} (2n-k-1) a_k^2$$

Throughout this part, $n$ denotes an integer greater than or equal to 3, and $p = [(n+1)/2]$ is the integer part of $(n+1)/2$. Let $a = (a_0, \ldots, a_{2n-2})$ be an element of $\mathbb{R}^{2n-1}$ and $M = H(a)$. We denote $\operatorname{Spo}(M) = (\lambda_1, \ldots, \lambda_n)$.
We define two vectors $v = (v_1, \ldots, v_n)$ and $w = (w_1, \ldots, w_n)$ of $\mathbb{R}^n$ by $$\begin{cases} v_i = \sqrt{2i-1}\, a_{2(i-1)} \text{ and } w_i = \dfrac{1}{\sqrt{2i-1}} & \text{if } i \in \{1, \ldots, p\} \\ v_i = \sqrt{2n-2i+1}\, a_{2(i-1)} \text{ and } w_i = \dfrac{1}{\sqrt{2n-2i+1}} & \text{if } i \in \{p+1, \ldots, n\} \end{cases}$$ Finally, we set $K_n = n - \|w\|^2$.
Show that $\langle v, w \rangle = \sum_{i=1}^{n} \lambda_i$ and $\|v\|^2 \leqslant \sum_{i=1}^{n} \lambda_i^2$.

Throughout this part, $n$ denotes an integer greater than or equal to 3, and $p = [(n+1)/2]$ is the integer part of $(n+1)/2$. Let $a = (a_0, \ldots, a_{2n-2})$ be an element of $\mathbb{R}^{2n-1}$ and $M = H(a)$. We denote $\operatorname{Spo}(M) = (\lambda_1, \ldots, \lambda_n)$.
We define two vectors $v = (v_1, \ldots, v_n)$ and $w = (w_1, \ldots, w_n)$ of $\mathbb{R}^n$ by $$\begin{cases} v_i = \sqrt{2i-1}\, a_{2(i-1)} \text{ and } w_i = \dfrac{1}{\sqrt{2i-1}} & \text{if } i \in \{1, \ldots, p\} \\ v_i = \sqrt{2n-2i+1}\, a_{2(i-1)} \text{ and } w_i = \dfrac{1}{\sqrt{2n-2i+1}} & \text{if } i \in \{p+1, \ldots, n\} \end{cases}$$ Finally, we set $K_n = n - \|w\|^2$.
Show that $\displaystyle\sum_{1 \leqslant i < j \leqslant n} (\lambda_i - \lambda_j)^2 = n \sum_{i=1}^{n} \lambda_i^2 - \langle v, w \rangle^2$ and deduce the inequality: $$\sum_{1 \leqslant i < j \leqslant n} (\lambda_i - \lambda_j)^2 \geqslant K_n \sum_{i=1}^{n} \lambda_i^2 \tag{III.1}$$

Throughout this part, $n$ denotes an integer greater than or equal to 3, and $p = [(n+1)/2]$ is the integer part of $(n+1)/2$.
Verify that if $n = 3$, condition III.1 is equivalent to: $2(\lambda_1^2 + \lambda_2^2 + \lambda_3^2) \geqslant 3(\lambda_1\lambda_2 + \lambda_1\lambda_3 + \lambda_2\lambda_3)$.

Throughout this part, $n$ denotes an integer greater than or equal to 3, and $p = [(n+1)/2]$ is the integer part of $(n+1)/2$.
Let $B = (b_{i,j})_{1 \leqslant i,j \leqslant n}$ be the matrix of $\mathcal{M}_n(\mathbb{R})$ defined by $$b_{1,2p-1} = 1 \quad b_{2p-1,1} = 1 \quad b_{p,p} = -2$$ all other coefficients of $B$ being zero.
Determine the ordered spectrum of matrix $B$.

Throughout this part, $n$ denotes an integer greater than or equal to 3, and $p = [(n+1)/2]$ is the integer part of $(n+1)/2$.
We admit the following result: if $A$ and $B$ are two matrices of $\mathcal{S}_n(\mathbb{R})$ whose respective eigenvalues (with possible repetitions) are $\alpha_1 \geqslant \ldots \geqslant \alpha_n$ and $\beta_1 \geqslant \ldots \geqslant \beta_n$ then $$\sum_{i=1}^{n} \alpha_i \beta_{n+1-i} \leqslant \operatorname{tr}(AB) \leqslant \sum_{i=1}^{n} \alpha_i \beta_i$$
Let $B = (b_{i,j})_{1 \leqslant i,j \leqslant n}$ be the matrix of $\mathcal{M}_n(\mathbb{R})$ defined by $$b_{1,2p-1} = 1 \quad b_{2p-1,1} = 1 \quad b_{p,p} = -2$$ all other coefficients of $B$ being zero.
Let $a = (a_0, \ldots, a_{2n-2})$ be an element of $\mathbb{R}^{2n-1}$ and $M = H(a)$. We denote $\operatorname{Spo}(M) = (\lambda_1, \ldots, \lambda_n)$.
Establish that $$\lambda_1 - \lambda_{n-1} - 2\lambda_n \geqslant 0 \quad \text{and} \quad 2\lambda_1 + \lambda_2 - \lambda_n \geqslant 0 \tag{III.3}$$

Throughout this part, $n$ denotes an integer greater than or equal to 3, and $p = [(n+1)/2]$ is the integer part of $(n+1)/2$.
Let $\lambda_1, \lambda_2, \lambda_3$ be three real numbers satisfying $$\lambda_1 \geqslant \lambda_2 \geqslant \lambda_3 \quad \lambda_1 - \lambda_2 - 2\lambda_3 \geqslant 0 \quad 2\lambda_1 + \lambda_2 - \lambda_3 \geqslant 0$$
We define the Hankel matrix $M = H(a,b,c,b,a) = \begin{pmatrix} a & b & c \\ b & c & b \\ c & b & a \end{pmatrix}$, where $a, b, c$ are real.
Calculate the eigenvalues of $M$ (without trying to order them).

Throughout this part, $n$ denotes an integer greater than or equal to 3, and $p = [(n+1)/2]$ is the integer part of $(n+1)/2$.
Let $\lambda_1, \lambda_2, \lambda_3$ be three real numbers satisfying $$\lambda_1 \geqslant \lambda_2 \geqslant \lambda_3 \quad \lambda_1 - \lambda_2 - 2\lambda_3 \geqslant 0 \quad 2\lambda_1 + \lambda_2 - \lambda_3 \geqslant 0$$
We define the Hankel matrix $M = H(a,b,c,b,a) = \begin{pmatrix} a & b & c \\ b & c & b \\ c & b & a \end{pmatrix}$, where $a, b, c$ are real.
Explicitly express $a, b, c$ (with $b \geqslant 0$) as functions of $\lambda_1, \lambda_2, \lambda_3$, such that $\operatorname{Spo}(M) = (\lambda_1, \lambda_2, \lambda_3)$.

Throughout this part, $n$ denotes an integer greater than or equal to 3, and $p = [(n+1)/2]$ is the integer part of $(n+1)/2$.
Let $\lambda_1, \lambda_2, \lambda_3$ be three real numbers satisfying $$\lambda_1 \geqslant \lambda_2 \geqslant \lambda_3 \quad \lambda_1 - \lambda_2 - 2\lambda_3 \geqslant 0 \quad 2\lambda_1 + \lambda_2 - \lambda_3 \geqslant 0 \tag{III.3}$$
We define the Hankel matrix $M = H(a,b,c,b,a) = \begin{pmatrix} a & b & c \\ b & c & b \\ c & b & a \end{pmatrix}$, where $a, b, c$ are real.
What can be deduced from the previous result, regarding condition III.3 in the case $n = 3$? Using an ordered triplet $(\lambda, 1, 1)$, show that for $n = 3$, condition III.1 is not sufficient.

Let $A = \left( a _ { i j } \right) _ { 1 \leqslant i , j \leqslant n } \in \mathcal { M } _ { n } ( \mathbb { R } )$. We define $R ( A ) = \left\{ { } ^ { t } X A X \mid X \in \mathbb { R } ^ { n } , \| X \| = 1 \right\}$.
Prove that the real eigenvalues of $A$ are in $R ( A )$.

Let $A = \left( a _ { i j } \right) _ { 1 \leqslant i , j \leqslant n } \in \mathcal { M } _ { n } ( \mathbb { R } )$. We define $R ( A ) = \left\{ { } ^ { t } X A X \mid X \in \mathbb { R } ^ { n } , \| X \| = 1 \right\}$.
I.B.1) Prove that the elements $a _ { i i } ( 1 \leqslant i \leqslant n )$ on the diagonal of $A$ are in $R ( A )$.
I.B.2) By considering the matrix $$A = \left( \begin{array} { c c } 0 & 1 \\ - 1 & 0 \end{array} \right)$$ show that the elements $a _ { i j }$ with $i \neq j$ are not necessarily in $R ( A )$.