We denote by $\operatorname { OSp } _ { n } ( \mathbb { R } ) = \operatorname { Sp } _ { n } ( \mathbb { R } ) \cap \mathrm { O } _ { n } ( \mathbb { R } )$ the set of real symplectic and orthogonal matrices, and $\mathcal { C } _ { J } = \left\{ M \in \mathcal { M } _ { n } ( \mathbb { R } ) \mid J M = M J \right\}$ the centralizer of $J$. Show that $\operatorname { OSp } _ { n } ( \mathbb { R } ) \subset \mathcal { C } _ { J }$.

We have $\operatorname { OSp } _ { n } ( \mathbb { R } ) \subset \mathcal { C } _ { J }$ and for every $M \in \mathcal{C}_J$, $\det(M) \geq 0$. Deduce that, for every matrix $M$ in $\operatorname { OSp } _ { n } ( \mathbb { R } ) , \operatorname { det } ( M ) = 1$.

Given that $\operatorname { OSp } _ { n } ( \mathbb { R } ) \subset \mathcal { C } _ { J }$ and that for every matrix $M \in \mathcal { C } _ { J }$, $\det(M) \geq 0$, deduce that, for every matrix $M$ in $\operatorname { OSp } _ { n } ( \mathbb { R } ) , \operatorname { det } ( M ) = 1$.

Let $M \in \mathcal{M}_{k \times d}(\mathbb{R})$ with $\operatorname{rank}(M) = k$, $b \in \mathbb{R}^k \backslash \{0\}$, and fix $\bar{x} \in C$ where $C := \{x \in \mathbb{R}^d : Mx = b, \|x\|_1 = r\}$. Let $q \in \operatorname{Ker}(M)^\perp \backslash \{0\}$ be as in question 23. Let $K$ be the set of $y \in \mathbb{R}^d$ such that $$My = b, \quad y_i = 0 \ \forall i \in I_0(\bar{x}), \quad q_i y_i \geqslant 0 \ \forall i \in \{1, \ldots, d\}$$ Show that $K$ is non-empty and included in $C$.

Let $M \in \mathcal{M}_{k \times d}(\mathbb{R})$ with $\operatorname{rank}(M) = k$, $b \in \mathbb{R}^k \setminus \{0\}$, $\bar{x} \in C$ (where $C := \{x \in \mathbb{R}^d : Mx = b, \|x\|_1 = r\}$), and $q \in \operatorname{Ker}(M)^\perp \setminus \{0\}$ as in question 23. Let $K$ be the set of $y \in \mathbb{R}^d$ such that $$My = b, \quad y_i = 0 \quad \forall i \in I_0(\bar{x}), \quad q_i y_i \geq 0 \quad \forall i \in \{1, \ldots, d\}.$$ Show that $K$ is non-empty and included in $C$.

Let $M \in \mathrm { Sp } _ { n } ( \mathbb { R } )$ and let $S \in \mathcal { S } _ { n } ( \mathbb { R } )$ be a symmetric matrix with strictly positive eigenvalues such that $S ^ { 2 } = M ^ { \top } M$. Show that $S$ is symplectic.
One may consider a basis of eigenvectors of the endomorphism $s$ of $\mathbb { R } ^ { n }$ canonically associated with $S$, and show that $s$ is a symplectic endomorphism of the standard space $(\mathbb { R } ^ { n } , b _ { s })$.

We consider a matrix $M \in \mathrm { Sp } _ { n } ( \mathbb { R } )$. Let $S \in \mathcal { S } _ { n } ( \mathbb { R } )$ be a symmetric matrix with strictly positive eigenvalues such that $S ^ { 2 } = M ^ { \top } M$. Show that $S$ is symplectic.
One may consider a basis of eigenvectors of the endomorphism $s$ of $\mathbb { R } ^ { n }$ canonically associated with $S$, and show that $s$ is a symplectic endomorphism of the standard space $(\mathbb { R } ^ { n } , b _ { s })$.

We consider $n$ discrete random variables $Y_1, \ldots, Y_n$ with random vector $Y$ and covariance matrix $\Sigma_Y$. The objective is to show that $\mathbb{P}\left(Y - \mathbb{E}(Y) \in \operatorname{Im}\Sigma_Y\right) = 1$. We now assume $r < n$ where $r$ is the rank of $\Sigma_Y$.
Prove that the kernel and image of $\Sigma_Y$ are supplementary orthogonal subspaces in $\mathcal{M}_{n,1}(\mathbb{R})$.

Let $M \in \mathcal{M}_{k \times d}(\mathbb{R})$ with $\operatorname{rank}(M) = k$, $b \in \mathbb{R}^k \backslash \{0\}$, and fix $\bar{x} \in C$ where $C := \{x \in \mathbb{R}^d : Mx = b, \|x\|_1 = r\}$. Let $K$ be as defined in question 24. Show that if $y \in \operatorname{Ext}(K)$ then $$h \in \operatorname{Ker}(M) \text{ and } I_0(y) \subset I_0(h) \Rightarrow h = 0.$$

With the notation of questions 23 and 24, show that if $y \in \operatorname{Ext}(K)$ then $$h \in \operatorname{Ker}(M) \text{ and } I_0(y) \subset I_0(h) \Rightarrow h = 0.$$

Let $M \in \mathrm { Sp } _ { n } ( \mathbb { R } )$ and let $S \in \mathcal { S } _ { n } ( \mathbb { R } )$ be a symmetric matrix with strictly positive eigenvalues such that $S ^ { 2 } = M ^ { \top } M$ and $S \in \mathrm{Sp}_n(\mathbb{R})$. Justify that $S$ is invertible then show that the matrix $O$ defined by $O = M S ^ { - 1 }$ belongs to the group $\mathrm { OSp } _ { n } ( \mathbb { R } )$.

We consider a matrix $M \in \mathrm { Sp } _ { n } ( \mathbb { R } )$ and $S \in \mathcal { S } _ { n } ( \mathbb { R } )$ a symmetric symplectic matrix with strictly positive eigenvalues such that $S ^ { 2 } = M ^ { \top } M$. Justify that $S$ is invertible then show that the matrix $O$ defined by $O = M S ^ { - 1 }$ belongs to the group $\mathrm { OSp } _ { n } ( \mathbb { R } )$.

Using the polar decomposition $M = OS$ where $O \in \operatorname{OSp}_n(\mathbb{R})$ and $S \in \mathrm{Sp}_n(\mathbb{R})$ is symmetric with strictly positive eigenvalues, conclude that the determinant of the matrix $M \in \mathrm{Sp}_n(\mathbb{R})$ is equal to 1.

Using the polar decomposition $M = OS$ where $O \in \operatorname{OSp}_n(\mathbb{R})$ and $S$ is a symmetric symplectic matrix with strictly positive eigenvalues, conclude that the determinant of the matrix $M \in \mathrm{Sp}_n(\mathbb{R})$ is equal to 1.

Let $(E , \omega)$ be a symplectic vector space of dimension $n = 2m$. Let $a \in E$ be a non-zero vector and $\lambda \in \mathbb { R }$ be a real number. Show that the map $\tau _ { a } ^ { \lambda }$ defined by
$$\forall x \in E , \quad \tau _ { a } ^ { \lambda } ( x ) = x + \lambda \omega ( a , x ) a$$
is a transvection of $E$ and that it is a symplectic endomorphism of this same space.

Let $(E, \omega)$ be a symplectic vector space of dimension $n = 2m$. Let $a \in E$ be a non-zero vector and $\lambda \in \mathbb { R }$ be a real number. Show that the map $\tau _ { a } ^ { \lambda }$ defined by
$$\forall x \in E , \quad \tau _ { a } ^ { \lambda } ( x ) = x + \lambda \omega ( a , x ) a$$
is a transvection of $E$ and that it is a symplectic endomorphism of this same space.

We assume that $\Sigma_Y$ satisfies $$\forall i \in \llbracket 1, n \rrbracket, \quad \sigma_{i,i} = \sigma^2 \quad \text{and} \quad \forall (i,j) \in \llbracket 1, n \rrbracket^2, \quad i \neq j \Longrightarrow \sigma_{i,j} = \sigma^2 \gamma$$ where $\sigma$ and $\gamma$ are two strictly positive real numbers. We denote by $J \in \mathcal{M}_n(\mathbb{R})$ the matrix whose coefficients are all equal to 1.
Prove that $\gamma \leqslant 1$ and express $\Sigma_Y$ in terms of $J$.

We assume that $\Sigma_Y$ satisfies $$\forall i \in \llbracket 1, n \rrbracket, \quad \sigma_{i,i} = \sigma^2 \quad \text{and} \quad \forall (i,j) \in \llbracket 1, n \rrbracket^2, \quad i \neq j \Longrightarrow \sigma_{i,j} = \sigma^2 \gamma$$ where $\sigma$ and $\gamma$ are two strictly positive real numbers. We denote by $J \in \mathcal{M}_n(\mathbb{R})$ the matrix whose coefficients are all equal to 1.
Determine the eigenvalues of $J$ and the dimension of each associated eigenspace. Also determine an eigenvector associated with its eigenvalue of maximal modulus.

We assume that $\Sigma_Y$ satisfies $$\forall i \in \llbracket 1, n \rrbracket, \quad \sigma_{i,i} = \sigma^2 \quad \text{and} \quad \forall (i,j) \in \llbracket 1, n \rrbracket^2, \quad i \neq j \Longrightarrow \sigma_{i,j} = \sigma^2 \gamma$$ where $\sigma$ and $\gamma$ are two strictly positive real numbers. We denote by $J \in \mathcal{M}_n(\mathbb{R})$ the matrix whose coefficients are all equal to 1.
Specify a unit vector $U_0$ such that the variance of $Z = U_0^\top Y$ is maximal.

We assume that $\Sigma_Y$ satisfies $$\forall i \in \llbracket 1, n \rrbracket, \quad \sigma_{i,i} = \sigma^2 \quad \text{and} \quad \forall (i,j) \in \llbracket 1, n \rrbracket^2, \quad i \neq j \Longrightarrow \sigma_{i,j} = \sigma^2 \gamma$$ where $\sigma$ and $\gamma$ are two strictly positive real numbers. We denote by $J \in \mathcal{M}_n(\mathbb{R})$ the matrix whose coefficients are all equal to 1, and $U_0$ is a unit vector such that the variance of $Z = U_0^\top Y$ is maximal.
Calculate the percentage of total variance represented by $Z$, that is, the ratio $\dfrac{\mathbb{V}(Z)}{\mathbb{V}_T(Y)}$.

Let $J \in M_{n}(\mathbb{C})$ be the matrix defined by
$$J = \left( \begin{array}{ccccc} 0 & 1 & 0 & \ldots & 0 \\ 0 & 0 & 1 & \cdots & 0 \\ \vdots & \ddots & \ddots & \ddots & \vdots \\ \vdots & & \ddots & 0 & 1 \\ 1 & 0 & \ldots & 0 & 0 \end{array} \right).$$
Calculate $J^{n}$ and show that $J$ is diagonalisable.

Let $J \in M_{n}(\mathbb{C})$ be the matrix defined by
$$J = \left( \begin{array}{ccccc} 0 & 1 & 0 & \ldots & 0 \\ 0 & 0 & 1 & \cdots & 0 \\ \vdots & \ddots & \ddots & \ddots & \vdots \\ \vdots & & \ddots & 0 & 1 \\ 1 & 0 & \ldots & 0 & 0 \end{array} \right).$$
Calculate the eigenvalues of $J$.

Let $J \in M_{n}(\mathbb{C})$ be the matrix defined by
$$J = \left( \begin{array}{ccccc} 0 & 1 & 0 & \ldots & 0 \\ 0 & 0 & 1 & \cdots & 0 \\ \vdots & \ddots & \ddots & \ddots & \vdots \\ \vdots & & \ddots & 0 & 1 \\ 1 & 0 & \ldots & 0 & 0 \end{array} \right).$$
Deduce the value of the determinant
$$\left| \begin{array}{ccccc} a_{0} & a_{1} & \ldots & a_{n-2} & a_{n-1} \\ a_{n-1} & a_{0} & \ddots & & a_{n-2} \\ \vdots & \ddots & \ddots & \ddots & \vdots \\ a_{2} & & \ddots & a_{0} & a_{1} \\ a_{1} & a_{2} & \cdots & a_{n-1} & a_{0} \end{array} \right|$$
where $a_{0}, \ldots, a_{n-1}$ are arbitrary complex numbers.

We denote $\| u \| = \sup _ { \substack { x \in E \\ x \neq 0 } } \frac { \| u ( x ) \| } { \| x \| }$. Show that $\|.\|$ is a norm on $\mathcal { L } ( E )$.

Show that $S_n^+(\mathrm{R})$ and $S_n^{++}(\mathrm{R})$ are convex subsets of $M_n(\mathrm{R})$. Are they vector subspaces of $M_n(\mathrm{R})$?