Let $d \in \llbracket 1 , n \rrbracket , \left( U _ { 1 } , \ldots , U _ { d } \right)$ be a linearly independent family in $\mathcal { M } _ { n , 1 } ( \mathbb { R } )$ and $H = \operatorname { Vect } \left( U _ { 1 } , \ldots , U _ { d } \right)$. Prove that there exist integers $i _ { 1 } , \ldots , i _ { d }$ satisfying $1 \leqslant i _ { 1 } < \cdots < i _ { d } \leqslant n$ such that the application $$\left\lvert \, \begin{array} { c c c }
H & \rightarrow & \mathcal { M } _ { d , 1 } ( \mathbb { R } ) \\
\left( \begin{array} { c }
x _ { 1 } \\
\vdots \\
x _ { n }
\end{array} \right) & \mapsto & \left( \begin{array} { c }
x _ { i _ { 1 } } \\
\vdots \\
x _ { i _ { d } }
\end{array} \right)
\end{array} \right.$$ is bijective. One may consider the rank of the matrix in $\mathcal { M } _ { n , d } ( \mathbb { R } )$ whose columns are $U _ { 1 } , \ldots , U _ { d }$.
Let $\mathcal { W }$ be a vector subspace of $\mathcal { M } _ { n , 1 } ( \mathbb { R } )$ of dimension $d$. Prove that $$\operatorname { card } \left( \mathcal { W } \cap \mathcal { V } _ { n , 1 } \right) \leqslant 2 ^ { d }$$
In this part, $\mathbf{K} = \mathbf{R}$. If $G$ is a closed subgroup of $\mathrm{GL}_n(\mathbf{R})$, we introduce its Lie algebra: $$\mathcal{A}_G = \left\{ M \in \mathcal{M}_n(\mathbf{R}) \mid \forall t \in \mathbf{R} \quad e^{tM} \in G \right\}.$$ The tangent space to $G$ at $I_n$ is denoted $\mathcal{T}_{I_n}(G)$. $\mathbf{17}$ ▷ Show that, in the particular cases $G = \mathrm{SL}_n(\mathbf{R})$ and $G = \mathrm{O}_n(\mathbf{R})$, we have $\mathcal{T}_{I_n}(G) = \mathcal{A}_G$.
We denote by $\mathcal { C } _ { J } = \left\{ M \in \mathcal { M } _ { n } ( \mathbb { R } ) \mid J M = M J \right\}$ the centralizer of the matrix $J = \left( \begin{array}{cc} 0 & -I_m \\ I_m & 0 \end{array} \right)$. Show that, for every matrix $M \in \mathcal { M } _ { n } ( \mathbb { R } ) \left( = \mathcal { M } _ { 2 m } ( \mathbb { R } ) \right)$, $$M \in \mathcal { C } _ { J } \quad \Longleftrightarrow \quad \exists ( U , V ) \in \mathcal { M } _ { m } ( \mathbb { R } ) \times \mathcal { M } _ { m } ( \mathbb { R } ) , \quad M = \left( \begin{array} { c c }
U & - V \\
V & U
\end{array} \right) .$$
We denote by $\mathcal { C } _ { J } = \left\{ M \in \mathcal { M } _ { n } ( \mathbb { R } ) \mid J M = M J \right\}$ the centralizer of $J = \left( \begin{array}{cc} 0 & -I_m \\ I_m & 0 \end{array} \right)$, and every $M \in \mathcal{C}_J$ has the form $M = \left( \begin{array}{cc} U & -V \\ V & U \end{array} \right)$ for some $U, V \in \mathcal{M}_m(\mathbb{R})$. Deduce that, for every matrix $M \in \mathcal { C } _ { J } , \operatorname { det } ( M ) \geqslant 0$. One may consider the product of block matrices $\left( \begin{array} { c c } I _ { m } & 0 \\ \mathrm { i } I _ { m } & I _ { m } \end{array} \right) \left( \begin{array} { c c } U & - V \\ V & U \end{array} \right) \left( \begin{array} { c c } I _ { m } & 0 \\ - \mathrm { i } I _ { m } & I _ { m } \end{array} \right)$.
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 in $\mathcal { M } _ { n } ( \mathbb { R } )$. We equip $\mathcal { M } _ { n } ( \mathbb { R } )$ with its topology as a normed vector space of finite dimension. Show that $\operatorname { OSp } _ { n } ( \mathbb { R } )$ is a compact subgroup of the symplectic group $\operatorname { Sp } _ { n } ( \mathbb { R } )$.
We denote by $\operatorname { OSp } _ { n } ( \mathbb { R } ) = \operatorname { Sp } _ { n } ( \mathbb { R } ) \cap \mathrm { O } _ { n } ( \mathbb { R } )$ and $\mathcal { C } _ { J } = \left\{ M \in \mathcal { M } _ { n } ( \mathbb { R } ) \mid J M = M J \right\}$. 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 $\operatorname { OSp } _ { n } ( \mathbb { R } ) , \operatorname { det } ( M ) = 1$.
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 $(\mathbb { R } ^ { n } , b _ { s })$.
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 $\mathrm { OSp } _ { n } ( \mathbb { R } )$.
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.
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 let $\lambda$ and $\mu$ be real numbers. The symplectic transvections are defined by $\tau _ { a } ^ { \lambda } ( x ) = x + \lambda \omega ( a , x ) a$. Show that $\tau _ { a } ^ { \mu } \circ \tau _ { a } ^ { \lambda } = \tau _ { a } ^ { \lambda + \mu }$.
Let $(E , \omega)$ be a symplectic vector space of dimension $n = 2m$. Let $a \in E$ be a non-zero vector and $\lambda$ be a real number. The symplectic transvection is defined by $\tau _ { a } ^ { \lambda } ( x ) = x + \lambda \omega ( a , x ) a$. Show that $\operatorname { det } \left( \tau _ { a } ^ { \lambda } \right) > 0$.
Let $(E , \omega)$ be a symplectic vector space of dimension $n = 2m$. Let $a \in E$ be a non-zero vector and $\lambda$ be a real number. The symplectic transvection is defined by $\tau _ { a } ^ { \lambda } ( x ) = x + \lambda \omega ( a , x ) a$. Is the inverse $\left( \tau _ { a } ^ { \lambda } \right) ^ { - 1 }$ still a symplectic transvection?
Let $(E , \omega)$ be a symplectic vector space of dimension $n = 2m$. We fix $x$ and $y$, non-zero, in $E$. Suppose that $\omega ( x , y ) \neq 0$. Show that there exists $\lambda \in \mathbb { R }$ such that $\tau _ { y - x } ^ { \lambda } ( x ) = y$, where $\tau_a^{\lambda}(x) = x + \lambda \omega(a,x)a$.
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 $\dfrac{\mathbb{V}(Z)}{\mathbb{V}_T(Y)}$.
We assume that $\Sigma_Y$ has $n$ distinct eigenvalues which we order in strictly decreasing order $\lambda_1 > \cdots > \lambda_n$. We equip ourselves with a vector $U_0$ such that $\mathbb{V}\left(U_0^\top Y\right) = \max_{U \in C} \mathbb{V}\left(U^\top Y\right)$, where $C = \left\{ U \in \mathcal{M}_{n,1}(\mathbb{R}) \mid U^\top U = 1 \right\}$. We denote $$C' = \left\{ U \in \mathcal{M}_{n,1}(\mathbb{R}) \mid U^\top U = 1 \text{ and } U_0^\top U = 0 \right\}.$$ Justify that $q_Y$ admits a maximum on $C'$.
We assume that $\Sigma_Y$ has $n$ distinct eigenvalues which we order in strictly decreasing order $\lambda_1 > \cdots > \lambda_n$. We equip ourselves with a vector $U_0$ such that $\mathbb{V}\left(U_0^\top Y\right) = \max_{U \in C} \mathbb{V}\left(U^\top Y\right)$, where $C = \left\{ U \in \mathcal{M}_{n,1}(\mathbb{R}) \mid U^\top U = 1 \right\}$. We denote $$C' = \left\{ U \in \mathcal{M}_{n,1}(\mathbb{R}) \mid U^\top U = 1 \text{ and } U_0^\top U = 0 \right\}.$$ Determine the value of $\max_{U \in C'} \mathbb{V}\left(U^\top Y\right)$ and specify a vector $U_1 \in C'$ such that $$\max_{U \in C'} \mathbb{V}\left(U^\top Y\right) = \mathbb{V}\left(U_1^\top Y\right).$$
We assume that $\Sigma_Y$ has $n$ distinct eigenvalues which we order in strictly decreasing order $\lambda_1 > \cdots > \lambda_n$. We equip ourselves with a vector $U_0$ such that $\mathbb{V}\left(U_0^\top Y\right) = \max_{U \in C} \mathbb{V}\left(U^\top Y\right)$, and a vector $U_1 \in C'$ such that $\mathbb{V}\left(U_1^\top Y\right) = \max_{U \in C'} \mathbb{V}\left(U^\top Y\right)$, where $$C' = \left\{ U \in \mathcal{M}_{n,1}(\mathbb{R}) \mid U^\top U = 1 \text{ and } U_0^\top U = 0 \right\}.$$ Calculate the covariance of the discrete random variables $U_0^\top Y$ and $U_1^\top Y$ (to simplify notation, one may assume $Y$ is centered, that is, $\mathbb{E}(Y) = 0$).
Use the results of subsection III.D to prove the inclusion $\mathrm { Sp } _ { n } ( \mathbb { R } ) \subset \mathrm { SL } _ { n } ( \mathbb { R } )$.
Show that a matrix $S \in S_n(\mathrm{R})$ belongs to $S_n^+(\mathrm{R})$ if, and only if, $\mathrm{Sp}(S) \subset \mathbf{R}_+$. Similarly, we will admit in the rest of the problem that: $S \in S_n^{++}(\mathrm{R})$ if, and only if, $\operatorname{Sp}(S) \subset \mathbf{R}_+^\star$.
a) Show that $\mathbb{H}$ is a sub-$\mathbb{R}$-algebra of $M_2(\mathbb{C})$ stable under $Z \mapsto Z^*$. b) Let $Z \in \mathbb{H}$. Calculate $ZZ^*$ and deduce that every non-zero element of $\mathbb{H}$ is invertible. c) Let $Z \in \mathbb{H}$. Show that $Z \in \mathbb{R}_{\mathbb{H}}$ if and only if $ZZ' = Z'Z$ for all $Z' \in \mathbb{H}$.
Let $\mathscr{P}$ be the set of row vectors of size $d$ with non-negative coefficients whose coordinate sum equals 1: $$\mathscr{P} = \left\{ u \in \mathscr{M}_{1,d}\left(\mathbb{R}_{+}\right) : \sum_{j=1}^{d} u_j = 1 \right\}.$$ We consider a square matrix $P \in \mathscr{M}_d\left(\mathbb{R}_{+}\right)$ such that for all $i \in \{1,\ldots,d\}$, $$\sum_{j=1}^{d} P_{i,j} = 1$$ We further assume that there exist $\nu \in \mathscr{P}$ and $c > 0$ such that for all $i,j \in \{1,\ldots,d\}$, $$P_{i,j} \geqslant c\nu_j.$$ Justify that $c \leqslant 1$.