Let $a_n,\ n \geq 0$ be complex numbers such that $\lim_n a_n = 0$.
(A) Show that $F(z) := \sum_{n \geq 0} a_n z^n$ is a holomorphic function on $\{z \in \mathbb{C} : |z| < 1\}$.
(B) Let $G(z)$ be a meromorphic function on $\{z \in \mathbb{C} : |z| < 2\}$, with a pole at 1. Show that $G \neq F$ on $\{z \in \mathbb{C} : |z| < 1\}$. (Hint: consider the function $(1-z)F(z)$ as $z \longrightarrow 1$.)

For the rest of this problem, we assume that $\varphi$ is a non-degenerate symmetric bilinear form on $E$, and we denote by $q$ its quadratic form.
Let $e = (e_1, \ldots, e_n)$ be a basis of $E$. We say that $e$ is $q$-orthogonal if and only if, for all $(i,j) \in \{1,\ldots,n\}^2$ with $i \neq j$, $\varphi(e_i, e_j) = 0$.
Suppose that $e$ is simultaneously $q$-orthogonal and $q'$-orthogonal. Show that, for all $i \in \{1,\ldots,n\}$, $e_i$ is an eigenvector of $h^{-1} \circ h'$.

We assume in this question that $\mathbb { K }$ is equal to $\mathbb { R }$. We equip the vector space $\mathbb { R } ^ { 3 }$ with its canonical Euclidean affine structure and its canonical frame. For every matrix $A$ of $\mathcal { M } _ { 0 } ( 2 , \mathbb { R } )$, we denote by $\mathcal { Q } _ { A }$ the set of points of $\mathbb { R } ^ { 3 }$ whose image by the application $j$ has the same characteristic polynomial as $A$.
a) Let $r$ be a strictly positive real number. Show that each of the sets $\mathcal { Q } _ { X _ { 0 } } , \mathcal { Q } _ { r J _ { 0 } }$ and $\mathcal { Q } _ { r H _ { 0 } }$ is a quadric for which an equation will be specified.
b) Draw graphically the appearance of the quadrics $\mathcal { Q } _ { X _ { 0 } } , \mathcal { Q } _ { J _ { 0 } }$ and $\mathcal { Q } _ { H _ { 0 } }$ on the same drawing.

Let $X, H, Y$ be three elements of $\mathcal { M } _ { 0 } ( 2 , \mathbb { K } )$ such that $(X, H, Y)$ forms an admissible triple. Fix a matrix $Q \in GL(2, \mathbb{K})$ satisfying $X = QX_0Q^{-1}$. We define the vectors $u = Q \binom{1}{0}$ and $v = Q \binom{0}{1}$.
a) By computing the vector $[ H , X ] u$ in two different ways, prove that $u$ is an eigenvector of the matrix $H$.
b) By computing the vector $[ H , X ] v$ in two different ways, prove the existence of a scalar $t$ satisfying the identity: $H = Q \left( \begin{array} { c c } 1 & t \\ 0 & - 1 \end{array} \right) Q ^ { - 1 }$.
c) Find a matrix $T \in GL ( 2 , \mathbb { K } )$ commuting with $X _ { 0 }$ and satisfying the relation $H = Q T H _ { 0 } ( Q T ) ^ { - 1 }$.

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 $\Psi(M)$.

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 $B = (b_{i,j})_{1 \leqslant i,j \leqslant n}$ 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 $\operatorname{Spo}(M) = (\lambda_1, \lambda_2, \lambda_3)$.

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 $R ( A ) = \left\{ { } ^ { t } X A X \mid X \in \mathbb { R } ^ { n } , \| X \| = 1 \right\}$.
Prove that if $\operatorname { Tr } ( A ) = 0$ then $0 \in R ( A )$.

Throughout this part $A$ and $B$ denote real symmetric matrices of $\mathcal { M } _ { 2 } ( \mathbb { R } )$. We denote by $\lambda _ { 1 } \leqslant \lambda _ { 2 }$ (resp. $\mu _ { 1 } \leqslant \mu _ { 2 }$) the eigenvalues of $A$ (resp. $B$). We define $R ( A ) = \left\{ { } ^ { t } X A X \mid X \in \mathbb { R } ^ { n } , \| X \| = 1 \right\}$.
Prove that $R ( A ) = \left[ \lambda _ { 1 } , \lambda _ { 2 } \right]$.

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 $\mathcal{S}_n^{++}(\mathbb{R})$.

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 $v^2 = u$.
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 ${}^t A A \in \mathcal{S}_n^{++}(\mathbb{R})$.

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

Let $A \in \mathcal{M}_n(\mathbb{R})$. We propose to give a necessary and sufficient condition for the existence of a solution $X \in \mathrm{GL}_n(\mathbb{R})$ to the system $$(*) : \left\{\begin{array}{l} {}^t A A + {}^t X X = I_n \\ {}^t A X - {}^t X A = 0_n \end{array}\right.$$
Show that if the system $(*)$ admits a solution in $\mathrm{GL}_n(\mathbb{R})$, then the eigenvalues of ${}^t A A$ belong to the interval $[0, 1[$.

For $p \in \mathbb{N}^*$, we set $$A_p = \left(\begin{array}{rrrrr} 2 & -1 & 0 & \cdots & 0 \\ -1 & 2 & -1 & \ddots & \vdots \\ 0 & -1 & 2 & \ddots & 0 \\ \vdots & \ddots & \ddots & \ddots & -1 \\ 0 & \cdots & 0 & -1 & 2 \end{array}\right) \in \mathcal{M}_p(\mathbb{R})$$ We denote by $P_p$ the polynomial such that, for all real $x$, $P_p(x) = \det(x I_p - A_p)$.
Determine the eigenvalues of $A_p$.

Let $f$ be a linear form on $\mathcal{M}_n(\mathbb{R})$, and let $A$ be the unique matrix in $\mathcal{M}_n(\mathbb{R})$ such that $\forall M \in \mathcal{M}_n(\mathbb{R}), f(M) = \operatorname{Tr}(AM)$.
Justify that ${}^t A A$ admits $n$ positive eigenvalues $\mu_1, \ldots, \mu_n$, counted with multiplicities.

In this question, $f$ denotes the linear form defined by $\forall M \in \mathcal{M}_n(\mathbb{R}), f(M) = \sum_{j=1}^n \sum_{i=j}^n m_{i,j}$, and $A$ is the matrix such that $\forall M \in \mathcal{M}_n(\mathbb{R}), f(M) = \operatorname{Tr}(AM)$, with $A^{-1}$ as given in IV.C.2.
Determine the eigenvalues of $A^{-1} {}^t A^{-1}$.