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 $F$ and $G$ be two vector subspaces of $E$.
a) Show that $(F+G)^\perp = F^\perp \cap G^\perp$.
b) Show that $(F \cap G)^\perp = F^\perp + G^\perp$.

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 $F$ be a vector subspace of $E$. We denote by $\varphi_F$ the restriction of $\varphi$ to $F^2$. We will say that $F$ is singular if and only if $\varphi_F$ is degenerate.
Show that $F$ is non-singular if and only if one of the following properties is verified:

• $F \cap F^\perp = \{0\}$;
• $E = F \oplus F^\perp$;
• $F^\perp$ is non-singular.

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.
We say that two vector subspaces $F$ and $G$ of $E$ are orthogonal if and only if for all $(x,y) \in F \times G$, $\varphi(x,y) = 0$.
If $F$ and $G$ are two vector subspaces of $E$ that are orthogonal and non-singular, show that $F \oplus G$ is non-singular.

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.
We assume that $E = \mathbb{R}^2$ and for all $(x,y) \in \mathbb{R}^2$, $q(x,y) = x^2 - y^2$ and $q'(x,y) = 2xy$.
Determine a $q$-orthogonal basis and a $q'$-orthogonal basis.

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.
With $q(x,y) = x^2 - y^2$ and $q'(x,y) = 2xy$ on $\mathbb{R}^2$ as defined in question II.B.1, does there exist a basis of $\mathbb{R}^2$ orthogonal for both $q$ and $q'$?

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 denote by $O(E,q)$ the set of isometries of $(E,q)$ into itself.
Let $f \in O(E,q)$. We assume that for all $x \in E$ such that $q(x) \neq 0$, we have $f(x) - x \neq 0$ and $q(f(x)-x) = 0$.
We propose to demonstrate that $f \in O^+(E,q)$ and that $E$ is an Artin space.
a) Show that $\operatorname{dim}(E) \geq 3$.
b) We denote by $V = \operatorname{Ker}(f - \operatorname{Id}_E)$. Show that $q_{/V} = 0$.
c) Let $x \in E$ such that $q(x) = 0$. We denote $H = \{x\}^\perp$. Show that $q_{/H}$ is not identically zero. Deduce that there exists $y \in E$ such that $q(x+y) = q(x-y) = q(y) \neq 0$.
d) We denote by $U = \operatorname{Im}(f - \operatorname{Id}_E)$. Show that $q_{/U} = 0$.
e) Show that $U^\perp = V = U$.
f) Deduce that $E$ is an Artin space and that $f \in O^+(E,q)$.

We wish to prove the Cartan-Dieudonné theorem, whose statement is: ``if $f \in O(E,q)$, $f$ is the composition of at most $n$ reflections, where $n = \operatorname{dim}(E)$, with the convention that $\operatorname{Id}_E$ is the composition of 0 reflections.''
Prove the Cartan-Dieudonné theorem when $n = 1$.

We wish to prove the Cartan-Dieudonné theorem, whose statement is: ``if $f \in O(E,q)$, $f$ is the composition of at most $n$ reflections, where $n = \operatorname{dim}(E)$, with the convention that $\operatorname{Id}_E$ is the composition of 0 reflections.''
We reason by induction, assuming $n > 1$ and that the Cartan-Dieudonné theorem is proved for any vector space of dimension $n-1$.
Conclude when there exists $x \in E$ such that $f(x) = x$ with $q(x) \neq 0$.

We wish to prove the Cartan-Dieudonné theorem, whose statement is: ``if $f \in O(E,q)$, $f$ is the composition of at most $n$ reflections, where $n = \operatorname{dim}(E)$, with the convention that $\operatorname{Id}_E$ is the composition of 0 reflections.''
We reason by induction, assuming $n > 1$ and that the Cartan-Dieudonné theorem is proved for any vector space of dimension $n-1$.
Conclude when there exists $x \in E$ such that $q(x) \neq 0$ and $q(f(x)-x) \neq 0$.

We wish to prove the Cartan-Dieudonné theorem, whose statement is: ``if $f \in O(E,q)$, $f$ is the composition of at most $n$ reflections, where $n = \operatorname{dim}(E)$, with the convention that $\operatorname{Id}_E$ is the composition of 0 reflections.''
We reason by induction, assuming $n > 1$ and that the Cartan-Dieudonné theorem is proved for any vector space of dimension $n-1$.
Conclude in the other cases (i.e., when neither of the conditions in IV.A.2 or IV.A.3 holds).

We propose to prove Witt's theorem, whose statement is: ``let $F$ and $F'$ be two vector subspaces of $E$ such that there exists an isometry $f$ from $(F, q_{/F})$ to $(F', q_{/F'})$. Then there exists $g \in O(E,q)$ such that $g_{/F} = f$.''
Show that we can reduce to the case where $F$ and $F'$ are non-singular.

We propose to prove Witt's theorem, whose statement is: ``let $F$ and $F'$ be two vector subspaces of $E$ such that there exists an isometry $f$ from $(F, q_{/F})$ to $(F', q_{/F'})$. Then there exists $g \in O(E,q)$ such that $g_{/F} = f$.''
We assume that $F$ and $F'$ are non-singular, with $\operatorname{dim}(F) = \operatorname{dim}(F') = 1$. Let $x \in F$ with $x \neq 0$. Set $y = f(x)$.
a) Show that $q(x+y)$ or $q(x-y)$ is non-zero.
b) Prove Witt's theorem in this case, using question III.A.2-d).

We propose to prove Witt's theorem, whose statement is: ``let $F$ and $F'$ be two vector subspaces of $E$ such that there exists an isometry $f$ from $(F, q_{/F})$ to $(F', q_{/F'})$. Then there exists $g \in O(E,q)$ such that $g_{/F} = f$.''
We now assume that $F$ and $F'$ are non-singular, with $\operatorname{dim}(F) = \operatorname{dim}(F') > 1$.
a) Show that there exist $F_1$ and $F_2$ non-singular, such that $F_1 \perp F_2$ and $F = F_1 \oplus F_2$, with $\operatorname{dim}(F_1) = \operatorname{dim}(F) - 1$.
b) Suppose that there exists $g \in O(E,q)$ such that $g_{/F_1} = f_{/F_1}$. Denote $F_1' = f(F_1)$. Show that $f(F_2) \subset F_1'^\perp$ and that $g(F_2) \subset F_1'^\perp$.
c) Show that there exists $$h \in O\left(F_1'^\perp, q_{/F_1'^\perp}\right) \text{ such that } h_{/g(F_2)} = (f \circ g^{-1})_{/g(F_2)}.$$
d) Show that there exists $k \in O(E,q)$ such that $k_{/F} = f$.

We propose to prove Witt's theorem, whose statement is: ``let $F$ and $F'$ be two vector subspaces of $E$ such that there exists an isometry $f$ from $(F, q_{/F})$ to $(F', q_{/F'})$. Then there exists $g \in O(E,q)$ such that $g_{/F} = f$.''
Prove Witt's theorem.

Let $\alpha$ be a non-zero element of $E$. Show, for every vector $x$ of $E$, the identity: $$\tau _ { \alpha } ( x ) = x - 2 \frac { \langle \alpha , x \rangle } { \langle \alpha , \alpha \rangle } \alpha$$

We denote by $(f_{n})_{n \geqslant 1}$ the sequence of functions defined on $]0, +\infty[$ by: $$f_{n}(t) = \begin{cases} \left(1 - \frac{t}{n}\right)^{n} t^{x-1} & \text{if } t \in ]0, n[ \\ 0 & \text{if } t \geqslant n \end{cases}$$ Show that for all integers $n$, $n \geqslant 1$, the function $f_{n}$ is continuous and integrable on $]0, +\infty[$.

Let $\varepsilon_{1}, \varepsilon_{2}, \varepsilon_{3}, \varepsilon_{4}$ be four strictly positive real numbers pairwise distinct and two strictly positive real numbers $E$ and $N$. Let $\Omega$ be the part, assumed to be non-empty, formed of the quadruplets $x = (x_{1}, x_{2}, x_{3}, x_{4})$ of $\mathbb{R}_{+}^{4}$ satisfying: $$\left\{\begin{array}{l} x_{1} + x_{2} + x_{3} + x_{4} = N \\ \varepsilon_{1} x_{1} + \varepsilon_{2} x_{2} + \varepsilon_{3} x_{3} + \varepsilon_{4} x_{4} = E \end{array}\right.$$
VI.A.1) Let $f$ be a function of class $\mathcal{C}^{1}$ on $\mathbb{R}_{+}^{4}$. Show that $f$ admits a maximum on $\Omega$. We then denote $a = (a_{1}, a_{2}, a_{3}, a_{4}) \in \Omega$ a point at which this maximum is attained.
VI.A.2) Show that if $(x_{1}, x_{2}, x_{3}, x_{4}) \in \Omega$ then $x_{3}$ and $x_{4}$ can be written in the form $$\begin{aligned} & x_{3} = u x_{1} + v x_{2} + w \\ & x_{4} = u^{\prime} x_{1} + v^{\prime} x_{2} + w^{\prime} \end{aligned}$$ where we shall give explicitly $u, v, u^{\prime}, v^{\prime}$ in terms of $\varepsilon_{1}, \varepsilon_{2}, \varepsilon_{3}, \varepsilon_{4}$.
VI.A.3) Assuming that none of the numbers $a_{1}, a_{2}, a_{3}, a_{4}$ is zero, deduce that $$\begin{aligned} & \frac{\partial f}{\partial x_{1}}(a) + u \frac{\partial f}{\partial x_{3}}(a) + u^{\prime} \frac{\partial f}{\partial x_{4}}(a) = 0 \\ & \frac{\partial f}{\partial x_{2}}(a) + v \frac{\partial f}{\partial x_{3}}(a) + v^{\prime} \frac{\partial f}{\partial x_{4}}(a) = 0 \end{aligned}$$
VI.A.4) Show that the vector subspace of $\mathbb{R}^{4}$ spanned by the vectors $(1, 0, u, u^{\prime})$ and $(0, 1, v, v^{\prime})$ admits a supplementary orthogonal subspace spanned by the vectors $(1,1,1,1)$ and $(\varepsilon_{1}, \varepsilon_{2}, \varepsilon_{3}, \varepsilon_{4})$.
VI.A.5) Deduce the existence of two real numbers $\alpha, \beta$ such that for all $i \in \{1,2,3,4\}$ we have $$\frac{\partial f}{\partial x_{i}}(a) = \alpha + \beta \varepsilon_{i}$$

Let $\varepsilon_{1}, \varepsilon_{2}, \varepsilon_{3}, \varepsilon_{4}$ be four strictly positive real numbers pairwise distinct and two strictly positive real numbers $E$ and $N$. Let $\Omega$ be the part, assumed to be non-empty, formed of the quadruplets $x = (x_{1}, x_{2}, x_{3}, x_{4})$ of $\mathbb{R}_{+}^{4}$ satisfying: $$\left\{\begin{array}{l} x_{1} + x_{2} + x_{3} + x_{4} = N \\ \varepsilon_{1} x_{1} + \varepsilon_{2} x_{2} + \varepsilon_{3} x_{3} + \varepsilon_{4} x_{4} = E \end{array}\right.$$ We define the function $F$ for all $x = (x_{1}, x_{2}, x_{3}, x_{4})$ of $\mathbb{R}_{+}^{4}$ by $$F(x_{1}, x_{2}, x_{3}, x_{4}) = -\sum_{i=1}^{4} \ln \Gamma(1 + x_{i})$$ We suppose that there exists $\bar{N} = (N_{1}, N_{2}, N_{3}, N_{4}) \in \Omega$, the numbers $N_{1}, N_{2}, N_{3}, N_{4}$ all being non-zero, such that $$\max_{x \in \Omega} F(x) = F(\bar{N})$$ Show the existence of two real numbers $\lambda$ and $\mu$ satisfying for all $i \in \{1,2,3,4\}$: $$\ln N_{i} + \frac{1}{2N_{i}} + \int_{0}^{+\infty} \frac{h(u)}{(u+N_{i})^{2}} du = \lambda + \mu \varepsilon_{i}$$

For all $i \in \{1,2,3,4\}$, we set $$\theta(N_{i}) = \frac{1}{2N_{i}} + \int_{0}^{+\infty} \frac{h(u)}{(u+N_{i})^{2}} du$$
VI.C.1) Show that for all $i \in \{1,2,3,4\}$ $$0 < \theta(N_{i}) < \frac{1}{N_{i}}$$
VI.C.2) Show the existence of a strictly positive real $K$ such that for all $i \in \{1,2,3,4\}$ $$N_{i} = K e^{\mu \varepsilon_{i}} e^{-\theta(N_{i})}$$

Write a procedure, in the Maple or Mathematica language, which takes as input a matrix $M \in \mathcal{S}_n(\mathbb{R})$ and which, using the characterization from I.B, returns ``true'' if the matrix $M$ is positive definite, and ``false'' otherwise.

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: $S_n = a_0^{(n)} + a_1^{(n)} X + \cdots + a_{n-1}^{(n)} X^{n-1}$. The family $\left(K_p\right)_{p \in \mathbb{N}}$ is the orthonormal family defined in question II.E.
Express $s_n$ using the sequence of polynomials $\left(K_p\right)_{p \in \mathbb{N}}$.

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: $$s_n = \sum_{1 \leqslant i,j \leqslant n} h_{i,j}^{(-1,n)}$$
Determine the value of $s_n$.

Let $k \in \mathbb { N } ^ { * }$. We denote by $c _ { k }$ the positive real number such that: $c _ { k } \int _ { 0 } ^ { 2 \pi } \left( \frac { 1 + \cos t } { 2 } \right) ^ { k } d t = 1$, and for every real $t$ we set $R _ { k } ( t ) = c _ { k } \left( \frac { 1 + \cos t } { 2 } \right) ^ { k }$.
Let $\epsilon \in ] 0 , \pi [$. We set: $d _ { k } ( \epsilon ) = \sup _ { t \in [ \epsilon , 2 \pi - \epsilon ] } R _ { k } ( t )$. Prove then that $$\lim _ { k \rightarrow + \infty } d _ { k } ( \epsilon ) = 0$$

Let $k \in \mathbb { N } ^ { * }$. We denote by $c _ { k }$ the positive real number such that: $c _ { k } \int _ { 0 } ^ { 2 \pi } \left( \frac { 1 + \cos t } { 2 } \right) ^ { k } d t = 1$, and for every real $t$ we set $R _ { k } ( t ) = c _ { k } \left( \frac { 1 + \cos t } { 2 } \right) ^ { k }$.
Let $\epsilon \in ] 0 , \pi [$, and $k \in \mathbb { N }$. Prove that for every $h \in C _ { 2 \pi } ( \mathbb { R } ; \mathbb { C } )$ that is of class $C ^ { 1 }$ on $\mathbb { R }$ and every real $u$, we have: $$\int _ { 0 } ^ { 2 \pi } R _ { k } ( u - t ) h ( t ) d t = \int _ { 0 } ^ { 2 \pi } R _ { k } \left( t _ { 1 } \right) h \left( u - t _ { 1 } \right) d t _ { 1 }$$ and $$\left| \int _ { 0 } ^ { 2 \pi } R _ { k } ( u - t ) h ( t ) d t - h ( u ) \right| \leq 2 \left\| h ^ { \prime } \right\| \epsilon + 4 \pi \| h \| d _ { k } ( \epsilon )$$ (We recall that $\int _ { 0 } ^ { 2 \pi } R _ { k } \left( t _ { 1 } \right) d t _ { 1 } = 1$ and that $\| h \|$ is defined at the beginning of the problem statement. To establish the inequality, one may use that $h \left( u - t _ { 1 } \right) = h \left( u - t _ { 1 } + 2 \pi \right)$ when $t _ { 1 } \in [ 2 \pi - \epsilon , 2 \pi ]$).