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 $X^* \in \mathcal{A}$ be such that $\alpha(X^*) = \sup_{X \in \mathcal{A}} \alpha(X) > 0$, and denote $\alpha^* = \alpha(X^*)$.
Show that:

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

Conclude that $c^* = \alpha^*$.

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 $c^* = \alpha^*$ be the minimal constant found previously, and $X^*$ the associated vector.
a) Show that $\Psi(M_{c^*}) X^* = 0$.
We set $Y^* = \frac{2}{c^*} \Psi(M) X^*$.
b) Show that the column vector $\binom{Y^*}{X^*}$ is an eigenvector of the $2n \times 2n$ matrix $\left(\begin{array}{cc}0 & 2\Psi(M) \\ -I_n & -4\Psi(D)\end{array}\right)$ and that $c^*$ is an eigenvalue of this matrix.

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 consider $\gamma$ a real eigenvalue of the matrix $\left(\begin{array}{cc}0 & 2\Psi(M) \\ -I_n & -4\Psi(D)\end{array}\right)$ and $\binom{X_1}{X_2}$ an associated eigenvector.
a) Show that ${}^t X_2 \Psi(M_\gamma) X_2 = 0$ and that $X_2 \neq 0$. Conclude that $\gamma \leqslant c^*$.
b) What conclusion do we draw from this on the calculation of the smallest additive constant $c^*$?

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 $R ( A )$.

Let $A = \left( a _ { i j } \right) _ { 1 \leqslant i , j \leqslant n } \in \mathcal { M } _ { n } ( \mathbb { R } )$. We define $R ( A ) = \left\{ { } ^ { t } X A X \mid X \in \mathbb { R } ^ { n } , \| X \| = 1 \right\}$.
We consider two real numbers $a \in R ( A )$ and $b \in R ( A )$, with $a < b$. Let $X _ { 1 }$ and $X _ { 2 }$ be two vectors of norm 1 such that ${ } ^ { t } X _ { 1 } A X _ { 1 } = a$, ${ } ^ { t } X _ { 2 } A X _ { 2 } = b$.
I.C.1) Prove that $X _ { 1 }$ and $X _ { 2 }$ are linearly independent.
I.C.2) We set $X _ { \lambda } = \lambda X _ { 1 } + ( 1 - \lambda ) X _ { 2 }$ for $0 \leqslant \lambda \leqslant 1$.
Prove that the function $\phi : \lambda \mapsto \frac { { } ^ { t } X _ { \lambda } A X _ { \lambda } } { \left\| X _ { \lambda } \right\| ^ { 2 } }$ is defined and continuous on the interval $[ 0,1 ]$.
I.C.3) Deduce that the segment $[ a , b ]$ is included in $R ( A )$.

Let $A = \left( a _ { i j } \right) _ { 1 \leqslant i , j \leqslant n } \in \mathcal { M } _ { n } ( \mathbb { R } )$. We define $R ( A ) = \left\{ { } ^ { t } X A X \mid X \in \mathbb { R } ^ { n } , \| X \| = 1 \right\}$.
Prove that if $\operatorname { Tr } ( A ) = 0$ then $0 \in R ( A )$.

In this part all matrices are of format $(n, n)$, where $n$ is an integer greater than or equal to 2. We say that a real symmetric matrix is positive definite if and only if all its eigenvalues are strictly positive.
Let $A$ and $B$ be two positive definite symmetric matrices, $\alpha$ and $\beta$ two real numbers $> 0$ such that $\alpha + \beta = 1$; prove that: $$\operatorname { det } ( \alpha A + \beta B ) \geqslant ( \operatorname { det } A ) ^ { \alpha } ( \operatorname { det } B ) ^ { \beta }$$

In this part all matrices are of format $(n, n)$, where $n$ is an integer greater than or equal to 2. We say that a real symmetric matrix is positive definite if and only if all its eigenvalues are strictly positive.
For $1 \leqslant i \leqslant k$, let $A _ { i }$ be positive definite symmetric matrices and $\alpha _ { i }$ strictly positive real numbers such that $\alpha _ { 1 } + \cdots + \alpha _ { k } = 1$. Prove that $$\operatorname { det } \left( \alpha _ { 1 } A _ { 1 } + \cdots + \alpha _ { k } A _ { k } \right) \geqslant \left( \operatorname { det } A _ { 1 } \right) ^ { \alpha _ { 1 } } \ldots \left( \operatorname { det } A _ { k } \right) ^ { \alpha _ { k } }$$
One may reason by induction on $k$.

Let $u$ be an endomorphism of $\mathbb{R}^n$. Show that $u$ is self-adjoint positive definite if and only if its matrix in any orthonormal basis belongs to $\mathcal{S}_n^{++}(\mathbb{R})$.

Show that if $S \in \mathcal{S}_n^{++}(\mathbb{R})$, then $S$ is invertible and $S^{-1} \in \mathcal{S}_n^{++}(\mathbb{R})$.

In this question, $u$ denotes an endomorphism of $\mathbb{R}^n$ that is self-adjoint positive definite. We propose to prove that there exists a unique endomorphism $v$ of $\mathbb{R}^n$ that is self-adjoint, positive definite, such that $v^2 = u$.
Let $v$ be an endomorphism of $\mathbb{R}^n$, self-adjoint positive definite and satisfying $v^2 = u$, and let $\lambda$ be an eigenvalue of $u$. Show that $v$ induces an endomorphism of $\operatorname{Ker}(u - \lambda \mathrm{Id})$ which we shall determine.

In this question, $u$ denotes an endomorphism of $\mathbb{R}^n$ that is self-adjoint positive definite. We propose to prove that there exists a unique endomorphism $v$ of $\mathbb{R}^n$ that is self-adjoint, positive definite, such that $v^2 = u$.
Deduce $v$, then conclude.

In this question, $u$ denotes an endomorphism of $\mathbb{R}^n$ that is self-adjoint positive definite. We propose to prove that there exists a unique endomorphism $v$ of $\mathbb{R}^n$ that is self-adjoint, positive definite, such that $v^2 = u$.
Show that there exists a polynomial $Q$ with real coefficients such that $v = Q(u)$.

Let $A \in \mathrm{GL}_n(\mathbb{R})$. Show that ${}^t A A \in \mathcal{S}_n^{++}(\mathbb{R})$.

Let $A \in \mathrm{GL}_n(\mathbb{R})$. Deduce that there exists a unique pair $(O, S) \in \mathrm{O}(n) \times \mathcal{S}_n^{++}(\mathbb{R})$ such that $A = OS$.

Let $A \in \mathrm{GL}_n(\mathbb{R})$. Determine the matrices $O$ and $S$ when $A = \left(\begin{array}{ccc} 3 & 0 & -1 \\ \sqrt{2}/2 & 3\sqrt{2} & -3\sqrt{2}/2 \\ -\sqrt{2}/2 & 3\sqrt{2} & 3\sqrt{2}/2 \end{array}\right)$.

Show that $\mathrm{O}(n)$ is a compact subset of $\mathcal{M}_n(\mathbb{R})$.

Show that $\mathcal{S}_n^+(\mathbb{R})$ is a closed subset of $\mathcal{M}_n(\mathbb{R})$.