We fix a basis $\mathbf{B}$ of $E$. We denote by $\mathcal{N}_{\mathbf{B}}$ the set of endomorphisms of $E$ whose matrix in $\mathbf{B}$ is strictly upper triangular. Justify that $\mathcal{N}_{\mathbf{B}}$ is a nilpotent vector subspace of $\mathcal{L}(E)$ and that its dimension equals $\frac{n(n-1)}{2}$.

In the case $n=1$: Let $M$ be an orthogonal matrix of size $2 \times 2$. We denote by $M_{1} = \binom{x_{1}}{x_{2}}$ and $M_{2} = \binom{y_{1}}{y_{2}}$ the two columns of $M$. Show the equivalence $$M \text{ is symplectic } \Longleftrightarrow M_{2} = -J_{1} M_{1}.$$

3a. We are given $q \in \mathbb { Q }$, $n \in \mathbb { N } ^ { \star }$ and a matrix $A \in S _ { n } ( \mathbb { Q } )$ such that $A ^ { 2 } = q I _ { n }$. Construct a matrix $B \in S _ { 2 n } ( \mathbb { Q } )$ commuting with the matrix $\left( \begin{array} { c c } A & 0 \\ 0 & A \end{array} \right)$ and such that $B ^ { 2 } = ( q + 1 ) I _ { 2 n }$.
3b. Show that for all $d \geqslant 1$, there exist $n \in \mathbb { N } ^ { \star }$ and matrices $M _ { 1 } , \ldots , M _ { d } \in S _ { n } ( \mathbb { Q } )$ that commute pairwise and such that $M _ { k } ^ { 2 } = k I _ { n }$ for all integers $1 \leqslant k \leqslant d$.
3c. Let $d \geqslant 1$ be an integer. Deduce that if $q _ { 1 } , \ldots , q _ { d } \in \mathbb { Q }$, $q _ { i } > 0$, then there exist $n \in \mathbb { N } ^ { \star }$ and matrices $M _ { 1 } , \ldots , M _ { d } \in S _ { n } ( \mathbb { Q } )$ that commute pairwise and such that $M _ { i } ^ { 2 } = q _ { i } I _ { n }$ for all $1 \leqslant i \leqslant d$.

3a. We are given $q \in \mathbb { Q } , n \in \mathbb { N } ^ { \star }$ and a matrix $A \in S _ { n } ( \mathbb { Q } )$ such that $A ^ { 2 } = q I _ { n }$. Construct a matrix $B \in S _ { 2 n } ( \mathbb { Q } )$ commuting with the matrix $\left( \begin{array} { c c } A & 0 \\ 0 & A \end{array} \right)$ and such that $B ^ { 2 } = ( q + 1 ) I _ { 2 n }$.
3b. Show that for all $d \geqslant 1$, there exist $n \in \mathbb { N } ^ { \star }$ and matrices $M _ { 1 } , \ldots , M _ { d } \in S _ { n } ( \mathbb { Q } )$ that commute pairwise and such that $M _ { k } ^ { 2 } = k I _ { n }$ for all integers $1 \leqslant k \leqslant d$.
3c. Let $d \geqslant 1$ be an integer. Deduce that if $q _ { 1 } , \ldots , q _ { d } \in \mathbb { Q } , q _ { i } > 0$, then there exist $n \in \mathbb { N } ^ { \star }$ and matrices $M _ { 1 } , \ldots , M _ { d } \in S _ { n } ( \mathbb { Q } )$ that commute pairwise and such that $M _ { i } ^ { 2 } = q _ { i } I _ { n }$ for all $1 \leqslant i \leqslant d$.

(a) Show that for all $u, v \in \mathbf{R}^p$, we have $\left(uu^T\right) \odot \left(vv^T\right) = (u \odot v)(u \odot v)^T$.
(b) Let $A \in \operatorname{Sym}^+(p)$. We denote $\lambda_1, \ldots, \lambda_p$ the eigenvalues (with multiplicity) of $A$ and $\left(u_1, \ldots, u_p\right)$ an orthonormal family of associated eigenvectors. Show that $\lambda_k \geq 0$ for all $k \in \llbracket 1, p \rrbracket$ and that $A = \sum_{k=1}^{p} \lambda_k u_k u_k^T$.
(c) Deduce that if $A, B \in \operatorname{Sym}^+(p)$ then $A \odot B \in \operatorname{Sym}^+(p)$.

Let $(E, \langle \cdot, \cdot \rangle)$ be a real pre-Hilbert space, with associated norm $\|\cdot\|$. Let $u$ be an endomorphism of $E$ satisfying, $$\forall (x,y) \in E^2, \quad \langle u(x), y \rangle = \langle x, u(y) \rangle$$ Suppose that there exists a unit vector $x_0 \in F$ satisfying $$\langle u(x_0), x_0 \rangle = \sup_{x \in F, \|x\|=1} \langle u(x), x \rangle$$ For every unit vector $y \in F$ orthogonal to $x_0$, we set, for every real $t$, $$\begin{aligned} & \gamma(t) = x_0 \cos t + y \sin t \\ & \varphi(t) = \langle u \circ \gamma(t), \gamma(t) \rangle \end{aligned}$$ Calculate $\|\gamma(t)\|$ then justify that $\varphi'(0) = 0$.

Let $(E, \langle \cdot, \cdot \rangle)$ be a real pre-Hilbert space, with associated norm $\|\cdot\|$. Let $u$ be an endomorphism of $E$ satisfying, $$\forall (x,y) \in E^2, \quad \langle u(x), y \rangle = \langle x, u(y) \rangle$$ Suppose that there exists a unit vector $x_0 \in F$ satisfying $$\langle u(x_0), x_0 \rangle = \sup_{x \in F, \|x\|=1} \langle u(x), x \rangle$$ For every unit vector $y \in F$ orthogonal to $x_0$, we set, for all real $t$, $$\begin{aligned} & \gamma(t) = x_0 \cos t + y \sin t \\ & \varphi(t) = \langle u \circ \gamma(t), \gamma(t) \rangle \end{aligned}$$ Calculate $\|\gamma(t)\|$ then justify that $\varphi'(0) = 0$.

Let $\mathbf{B}$ be a basis of $E$. Show that
$$\left\{\nu(u) \mid u \in \mathcal{N}_{\mathbf{B}}\right\} = \{\nu(u) \mid u \in \mathcal{N}(E)\} = \llbracket 1, n \rrbracket$$

In the case $n=1$: Let $X_{1} \in \mathcal{M}_{2,1}(\mathbb{R})$ have norm 1. Show that the square matrix consisting of columns $X_{1}$ and $-J_{1} X_{1}$ is both orthogonal and symplectic.

Let $n \in \mathbf{N}$ and $P : \mathbf{R} \rightarrow \mathbf{R}$ defined by $P(x) = \sum_{k=0}^{n} a_k x^k$ where $a_k \geq 0$ for all $k \in \llbracket 0, n \rrbracket$ a polynomial with non-negative coefficients.
(a) Verify that $P[A] = \sum_{k=0}^{n} a_k A^{(k)}$ for all matrices $A \in \mathcal{M}_p(\mathbf{R})$.
(b) Show that if $A \in \operatorname{Sym}^+(p)$ then $P[A] \in \operatorname{Sym}^+(p)$.

Let $(E, \langle \cdot, \cdot \rangle)$ be a real pre-Hilbert space, with associated norm $\|\cdot\|$. Let $u$ be an endomorphism of $E$ satisfying, $$\forall (x,y) \in E^2, \quad \langle u(x), y \rangle = \langle x, u(y) \rangle$$ Suppose that there exists a unit vector $x_0 \in F$ satisfying $$\langle u(x_0), x_0 \rangle = \sup_{x \in F, \|x\|=1} \langle u(x), x \rangle$$ For every unit vector $y \in F$ orthogonal to $x_0$, we set, for every real $t$, $$\begin{aligned} & \gamma(t) = x_0 \cos t + y \sin t \\ & \varphi(t) = \langle u \circ \gamma(t), \gamma(t) \rangle \end{aligned}$$ Deduce that $u(x_0)$ is orthogonal to $y$.

Let $(E, \langle \cdot, \cdot \rangle)$ be a real pre-Hilbert space, with associated norm $\|\cdot\|$. Let $u$ be an endomorphism of $E$ satisfying, $$\forall (x,y) \in E^2, \quad \langle u(x), y \rangle = \langle x, u(y) \rangle$$ Suppose that there exists a unit vector $x_0 \in F$ satisfying $$\langle u(x_0), x_0 \rangle = \sup_{x \in F, \|x\|=1} \langle u(x), x \rangle$$ For every unit vector $y \in F$ orthogonal to $x_0$, we set, for all real $t$, $$\begin{aligned} & \gamma(t) = x_0 \cos t + y \sin t \\ & \varphi(t) = \langle u \circ \gamma(t), \gamma(t) \rangle \end{aligned}$$ Deduce that $u(x_0)$ is orthogonal to $y$.

Let $u \in \mathcal{L}(E)$. We are given two vectors $x$ and $y$ of $E$, as well as two integers $p \geq q \geq 1$ such that $u^{p}(x) = u^{q}(y) = 0$ and $u^{p-1}(x) \neq 0$. Show that the family $(x, u(x), \ldots, u^{p-1}(x))$ is free, and that if $(u^{p-1}(x), u^{q-1}(y))$ is free then $(x, u(x), \ldots, u^{p-1}(x), y, u(y), \ldots, u^{q-1}(y))$ is free.

In the case $n=1$: Let $M$ be a matrix of size $2 \times 2$ that is symmetric and symplectic. Show that $M$ is diagonalizable and that its eigenvalues are inverses of each other. Show that there exists a matrix $P$ that is both orthogonal and symplectic such that $P^{-1} M P$ is diagonal.

We set, for all $n \geq 0$ and all $x \in \mathbf{R}$, $P_n(x) = \sum_{k=0}^{n} \frac{x^k}{k!}$ where $k!$ denotes the factorial of $k$.
Let $A \in \operatorname{Sym}^+(p)$.
(a) Show that for all $(i,j) \in \llbracket 1,p \rrbracket^2$, we have $$\lim_{n \rightarrow +\infty} P_n[A]_{ij} = \exp\left(A_{ij}\right)$$
(b) Show that $\exp[A] \in \operatorname{Sym}^+(p)$.
(c) Let $u \in \mathbf{R}^p$. Show that $\exp[A] \odot \left(uu^T\right) \in \operatorname{Sym}^+(p)$.

Let $(E, \langle \cdot, \cdot \rangle)$ be a real pre-Hilbert space, with associated norm $\|\cdot\|$. Let $u$ be an endomorphism of $E$ satisfying, $$\forall (x,y) \in E^2, \quad \langle u(x), y \rangle = \langle x, u(y) \rangle$$ Suppose that there exists a unit vector $x_0 \in F$ satisfying $$\langle u(x_0), x_0 \rangle = \sup_{x \in F, \|x\|=1} \langle u(x), x \rangle$$ Show that $x_0$ is an eigenvector of $u$.

Let $(E, \langle \cdot, \cdot \rangle)$ be a real pre-Hilbert space, with associated norm $\|\cdot\|$. Let $u$ be an endomorphism of $E$ satisfying, $$\forall (x,y) \in E^2, \quad \langle u(x), y \rangle = \langle x, u(y) \rangle$$ Suppose that there exists a unit vector $x_0 \in F$ satisfying $$\langle u(x_0), x_0 \rangle = \sup_{x \in F, \|x\|=1} \langle u(x), x \rangle$$ Show that $x_0$ is an eigenvector of $u$.

Let $u \in \mathcal{N}(E)$, with nilindex $p$. Deduce from the previous question that if $p \geq n-1$ and $p \geq 2$ then $\operatorname{Im} u^{p-1} = \operatorname{Im} u \cap \operatorname{Ker} u$ and $\operatorname{Im} u^{p-1}$ has dimension 1.

In the case $n=1$: Determine the matrices of size $2 \times 2$ that are both antisymmetric and symplectic and show that they are not diagonalizable in $\mathbb{R}$.

Let $d \in \mathbf{N}_*$. We consider a $p$-tuple $\left(x_i\right)_{1 \leq i \leq p}$ of elements of $\mathbf{R}^d$ and the matrix $$A = \left(\left\langle x_i, x_j \right\rangle\right)_{(i,j) \in \llbracket 1,p \rrbracket^2}$$ where $\langle a, b \rangle$ denotes the usual inner product between two vectors $a$ and $b$ of $\mathbf{R}^d$. We denote $|a| = \sqrt{\langle a, a \rangle}$ the norm of $a$.
(a) Show that $A \in \operatorname{Sym}^+(p)$.
(b) We denote $u \in \mathbf{R}^p$ the vector with coordinates $\left(\exp\left(-\frac{|x_1|^2}{2}\right), \ldots, \exp\left(-\frac{|x_p|^2}{2}\right)\right)$. Show that $\left(\exp[A] \odot \left(uu^T\right)\right)_{ij} = \exp\left(-\frac{|x_i - x_j|^2}{2}\right)$ for all $(i,j) \in \llbracket 1,p \rrbracket^2$.
(c) Let $\lambda > 0$ and $K \in \mathcal{M}_p(\mathbf{R})$ the matrix defined by $K_{ij} = \exp\left(-\frac{|x_i - x_j|^2}{2\lambda}\right)$ for all $(i,j) \in \llbracket 1,p \rrbracket^2$. Show that $K \in \operatorname{Sym}^+(p)$.

We consider a Euclidean vector space $(E, (-\mid-))$. Given $a \in E$ and $x \in E$, we denote by $a \otimes x$ the map from $E$ to itself defined by:
$$\forall z \in E, (a \otimes x)(z) = (a \mid z) \cdot x$$
We fix $x \in E \backslash \{0\}$. Show that the map $a \in E \mapsto a \otimes x$ is linear and constitutes a bijection from $E$ onto $\{u \in \mathcal{L}(E) : \operatorname{Im} u \subset \operatorname{Vect}(x)\}$.

Let $K$ be an antisymmetric matrix and $\varphi$ the application from $\left(\mathcal{M}_{2n,1}(\mathbb{R})\right)^{2}$ to $\mathbb{R}$ such that $$\forall (X,Y) \in \left(\mathcal{M}_{2n,1}(\mathbb{R})\right)^{2}, \quad \varphi(X,Y) = X^{\top} K Y.$$ Show that $\varphi$ is a bilinear form on $\mathcal{M}_{2n,1}(\mathbb{R})$.

We consider a Euclidean vector space $(E, (-\mid-))$. Given $a \in E$ and $x \in E$, we denote by $a \otimes x$ the map from $E$ to itself defined by:
$$\forall z \in E, (a \otimes x)(z) = (a \mid z) \cdot x$$
Let $a \in E$ and $x \in E \backslash \{0\}$. Show that $\operatorname{tr}(a \otimes x) = (a \mid x)$.

Let $K$ be an antisymmetric matrix and $\varphi$ the application from $\left(\mathcal{M}_{2n,1}(\mathbb{R})\right)^{2}$ to $\mathbb{R}$ such that $$\forall (X,Y) \in \left(\mathcal{M}_{2n,1}(\mathbb{R})\right)^{2}, \quad \varphi(X,Y) = X^{\top} K Y.$$ By computing $\varphi(X,X)^{\top}$ in two ways, show that $\varphi$ is alternating. Show similarly that $\varphi$ is antisymmetric.

We consider an $\mathbf{R}$-vector space $E$ of dimension $n > 0$. Let $\mathcal{V}$ be a nilpotent vector subspace of $\mathcal{L}(E)$ containing a non-zero element, with generic nilindex $p := \max_{u \in \mathcal{V}} \nu(u)$. In questions 8 to 11, we are given two arbitrary elements $u$ and $v$ of $\mathcal{V}$.
Let $k \in \mathbf{N}^{*}$. Show that there exists a unique family $(f_{0}^{(k)}, \ldots, f_{k}^{(k)})$ of endomorphisms of $E$ such that
$$\forall t \in \mathbf{R}, (u + tv)^{k} = \sum_{i=0}^{k} t^{i} f_{i}^{(k)}$$
Show in particular that $f_{0}^{(k)} = u^{k}$ and $f_{1}^{(k)} = \sum_{i=0}^{k-1} u^{i} v u^{k-1-i}$.