Show that for all matrices $A$ and $B$ in $\operatorname{Sym}^+(p)$ and all non-negative real numbers $a$ and $b$, we have $aA + bB \in \operatorname{Sym}^+(p)$.

In the case $n=1$: Show that a matrix of size $2 \times 2$ is symplectic if and only if its determinant equals 1.
Recall: $J_{1} = \left(\begin{array}{cc} 0 & 1 \\ -1 & 0 \end{array}\right)$, and a matrix $M \in \mathcal{M}_{2}(\mathbb{R})$ is symplectic if and only if $M^{\top} J_{1} M = J_{1}$.

Show that if $v \in \mathbf{R}^p$ then the matrix $A = \left(A_{ij}\right)_{(i,j) \in \llbracket 1,p \rrbracket^2}$ defined by $A = vv^T$ is in $\mathrm{Sym}^+(p)$.

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 $A \odot B \in \operatorname{Sym}^+(p)$.

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 $P[A] \in \operatorname{Sym}^+(p)$.

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 $d \in \mathbf{N}_*$. 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 $K \in \operatorname{Sym}^+(p)$.

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 $\mathcal{M}_{2n,1}(\mathbb{R})$.

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.

Throughout the rest of the problem, $K = J_{n}$. For all $X = \left(\begin{array}{c} x_{1} \\ x_{2} \\ \vdots \\ x_{2n} \end{array}\right) \in \mathcal{M}_{2n,1}(\mathbb{R})$ and for all $Y = \left(\begin{array}{c} y_{1} \\ y_{2} \\ \vdots \\ y_{2n} \end{array}\right) \in \mathcal{M}_{2n,1}(\mathbb{R})$, where $\varphi(X,Y) = X^{\top} J_{n} Y$, show the equality $$\varphi(X,Y) = \sum_{k=1}^{n} \left(x_{k} y_{k+n} - x_{k+n} y_{k}\right).$$

With $K = J_{n}$ and $\varphi(X,Y) = X^{\top} J_{n} Y$: Show that for all $(i,j) \in \{1,\ldots,2n\}^{2}$, $$\varphi(e_{i}, e_{j}) = \delta_{i+n,j} - \delta_{i,j+n}$$ (one may start with the case where $(i,j) \in \{1,\ldots,n\}^{2}$ then generalize).

Throughout part II, $A \in \mathcal{M}_n(\mathbb{R})$ is a strictly positive matrix. Show that, for all $x \in \mathbb{R}^n$, $$\left\{\begin{array}{l} x \geqslant 0 \Longrightarrow Ax \geqslant 0 \\ x \geqslant 0 \text{ and } x \neq 0 \Longrightarrow Ax > 0. \end{array}\right.$$

With $K = J_{n}$ and $\varphi(X,Y) = X^{\top} J_{n} Y$: Show that for all $X \in \mathcal{M}_{2n,1}(\mathbb{R})$, $J_{n} X \in X^{\perp}$ and compute $\varphi(J_{n} X, X)$.

Throughout part II, $A \in \mathcal{M}_n(\mathbb{R})$ is a strictly positive matrix. Show that $\forall k \in \mathbb{N}^*, A^k > 0$.

With $K = J_{n}$ and $\varphi(X,Y) = X^{\top} J_{n} Y$: If $Y \in \mathcal{M}_{2n,1}(\mathbb{R})$, we denote by $Y^{J_{n}}$ the set of vectors $Z$ of $\mathcal{M}_{2n,1}(\mathbb{R})$ such that $\varphi(Y,Z) = 0$. Show that $X^{J_{n}} = (J_{n} X)^{\perp}$.

Throughout part II, $A \in \mathcal{M}_n(\mathbb{R})$ is a strictly positive matrix. Deduce that $\rho(A) > 0$ then show that $\rho\left(\frac{A}{\rho(A)}\right) = 1$.

With $K = J_{n}$ and $\varphi(X,Y) = X^{\top} J_{n} Y$: Let $P$ be a symplectic and orthogonal matrix whose columns are denoted $X_{1}, \ldots, X_{2n}$. Show that, for all $(i,j) \in \{1,\ldots,2n\}^{2}$, $$\left\{\begin{array}{l} \|X_{i}\| = 1 \\ i \neq j \Longrightarrow X_{i} \perp X_{j} \\ \varphi(X_{i}, X_{j}) = \delta_{i+n,j} - \delta_{i,j+n} \end{array}\right.$$

We fix a real vector space $E$ of dimension $n$, as well as a nilpotent vector subspace $\mathcal{V}$ of $\mathcal{L}(E)$, equipped with an inner product $(-\mid-)$. We consider an arbitrary vector $x$ of $E \backslash \{0\}$, and set
$$H := \operatorname{Vect}(x)^{\perp}, \quad \mathcal{V} x := \{v(x) \mid v \in \mathcal{V}\} \text{ and } \mathcal{W} := \{v \in \mathcal{V} : v(x) = 0\}$$
We denote by $\pi$ the orthogonal projection of $E$ onto $H$. For $u \in \mathcal{W}$, we denote by $\bar{u}$ the endomorphism of $H$ defined by $\forall z \in H, \bar{u}(z) = \pi(u(z))$. We consider the sets $\overline{\mathcal{V}} := \{\bar{u} \mid u \in \mathcal{W}\}$ and $\mathcal{Z} := \{u \in \mathcal{W} : \bar{u} = 0\}$.
Show that $\mathcal{V} x$, $\mathcal{W}$, $\overline{\mathcal{V}}$ and $\mathcal{Z}$ are vector subspaces of $E$, $\mathcal{V}$, $\mathcal{L}(H)$ and $\mathcal{V}$ respectively.

Throughout part II, $A \in \mathcal{M}_n(\mathbb{R})$ is a strictly positive matrix. Assume $A$ is diagonalizable over $\mathbb{C}$. Show that, if $\rho(A) < 1$ then $\lim_{k \rightarrow +\infty} A^k = 0$.

With $K = J_{n}$, $\varphi(X,Y) = X^{\top} J_{n} Y$, and $P$ a symplectic and orthogonal matrix with columns $X_{1}, \ldots, X_{2n}$ satisfying the properties of Q13: Show that, for all $i \in \{1,\ldots,n\}$, $X_{i}^{J_{n}} = X_{i+n}^{\perp}$.

We fix a real vector space $E$ of dimension $n$, as well as a nilpotent vector subspace $\mathcal{V}$ of $\mathcal{L}(E)$, equipped with an inner product $(-\mid-)$. We consider an arbitrary vector $x$ of $E \backslash \{0\}$, and set
$$H := \operatorname{Vect}(x)^{\perp}, \quad \mathcal{V} x := \{v(x) \mid v \in \mathcal{V}\} \text{ and } \mathcal{W} := \{v \in \mathcal{V} : v(x) = 0\}$$
We denote by $\pi$ the orthogonal projection of $E$ onto $H$. For $u \in \mathcal{W}$, we denote by $\bar{u}$ the endomorphism of $H$ defined by $\forall z \in H, \bar{u}(z) = \pi(u(z))$. We consider the sets $\overline{\mathcal{V}} := \{\bar{u} \mid u \in \mathcal{W}\}$ and $\mathcal{Z} := \{u \in \mathcal{W} : \bar{u} = 0\}$.
Show that
$$\operatorname{dim} \mathcal{V} = \operatorname{dim}(\mathcal{V} x) + \operatorname{dim} \mathcal{Z} + \operatorname{dim} \overline{\mathcal{V}}.$$