Let $$A = \frac{1}{8} \left(\begin{array}{llll} 9 & 1 & 3 & 3 \\ 1 & 9 & 3 & 3 \\ 3 & 3 & 9 & 1 \\ 3 & 3 & 1 & 9 \end{array}\right).$$ Construct an orthogonal and symplectic matrix $P$ such that $P^{\top} A P$ is diagonal.

Let $M \in \mathcal{A}_{2n}(\mathbb{R}) \cap \mathrm{Sp}_{2n}(\mathbb{R})$, and let $m$ be the linear map canonically associated with $M$. Let $X$ denote an eigenvector of $M^{2}$ of norm 1 associated with a certain eigenvalue $\lambda$, and let $F = \operatorname{Vect}(X, MX, J_{n} X, J_{n} MX)$. Justify that if $\lambda \neq -1$, $F$ is a vector space of dimension 4. Show that, in this case, $$\left(X,\ \frac{-1}{\sqrt{-\lambda}} MX,\ -J_{n} X,\ \frac{1}{\sqrt{-\lambda}} J_{n} MX\right)$$ is an orthonormal basis of $F$. Then give the matrix of the application $m_{F}$ induced by $m$ on $F$ in the basis obtained.

Let $$B = \frac{1}{4} \left(\begin{array}{cccc} 0 & -5 & 0 & -3 \\ 5 & 0 & 3 & 0 \\ 0 & -3 & 0 & -5 \\ 3 & 0 & 5 & 0 \end{array}\right).$$ Determine a real number $a$ and a matrix $P$ such that $$P \in \mathcal{O}_{4}(\mathbb{R}) \cap \mathrm{Sp}_{4}(\mathbb{R}) \quad \text{and} \quad P^{\top} B P = \left(\begin{array}{cccc} 0 & a & 0 & 0 \\ -a & 0 & 0 & 0 \\ 0 & 0 & 0 & 1/a \\ 0 & 0 & -1/a & 0 \end{array}\right).$$

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

We consider the directed graph $G = ( S , A )$ where $$\left\{ \begin{array} { l } S = \{ 1,2,3,4 \} \\ A = \{ ( 1,2 ) , ( 2,1 ) , ( 1,3 ) , ( 3,1 ) , ( 1,4 ) , ( 4,1 ) , ( 2,3 ) , ( 3,2 ) , ( 2,4 ) , ( 4,2 ) , ( 3,4 ) , ( 4,3 ) \} \end{array} \right.$$ We assume that, when the point is on one of the vertices of the graph, it has the same probability of going to each of the three other vertices of the graph. We set $$J _ { 4 } = \left( \begin{array} { l l l l } 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 \end{array} \right)$$ Prove that there exists a matrix $Q \in \mathcal { O } _ { 4 } ( \mathbb { R } )$ such that $$T = \frac { 1 } { 3 } Q \left( \begin{array} { c c c c } - 1 & 0 & 0 & 0 \\ 0 & - 1 & 0 & 0 \\ 0 & 0 & - 1 & 0 \\ 0 & 0 & 0 & 3 \end{array} \right) Q ^ { \top }$$

We consider $n$ discrete random variables $Y_1, \ldots, Y_n$ with random vector $Y$ and covariance matrix $\Sigma_Y$. We denote by $P$ the change of basis matrix from the canonical basis of $\mathcal{M}_{n,1}(\mathbb{R})$ to an orthonormal basis formed by eigenvectors of $\Sigma_Y$. We define the discrete random variable $X = P^\top Y = \left(\begin{array}{c} X_1 \\ \vdots \\ X_n \end{array}\right)$.
Prove that $\Sigma_X$ is a diagonal matrix.

Show that, if $A \in S_n^{++}(\mathrm{R})$, there exists $S \in S_n^{++}(\mathrm{R})$ such that $A = S^2$.

Let $A \in S_n^{++}(\mathbf{R})$ and $B \in S_n(\mathbf{R})$. Show that there exists a diagonal matrix $D \in M_n(\mathbf{R})$ and $Q \in GL_n(\mathbf{R})$ such that $B = QDQ^\top$ and $A = QQ^\top$. What can be said about the diagonal elements of $D$ if $B \in S_n^{++}(\mathbf{R})$?
Hint: You may use question 3.

Show that, if $A \in S _ { n } ^ { + + } ( \mathbf { R } )$, there exists $S \in S _ { n } ^ { + + } ( \mathbf { R } )$ such that $A = S ^ { 2 }$.

Let $A \in S _ { n } ^ { + + } ( \mathbf { R } )$ and $B \in S _ { n } ( \mathbf { R } )$. Show that there exists a diagonal matrix $D \in M _ { n } ( \mathbf { R } )$ and $Q \in G L _ { n } ( \mathbf { R } )$ such that $B = Q D Q ^ { \top }$ and $A = Q Q ^ { \top }$. What can be said about the diagonal elements of $D$ if $B \in S _ { n } ^ { + + } ( \mathbf { R } )$ ?
Hint: You may use question 3.

Let $M$ be a symmetric matrix of $\mathcal{M}_n(\mathbf{R})$. Show that if $M$ is positive, then there exists $B \in \mathcal{M}_n(\mathbf{R})$ such that $M = B^T \cdot B$. Deduce that if $M$ is no longer assumed to be positive, but admits a unique strictly positive eigenvalue $\lambda$ with eigenspace of dimension 1 and unit eigenvector $u$, then there exists $B \in \mathcal{M}_n(\mathbf{R})$ such that $M = \lambda u \cdot u^T - B^T \cdot B$.

We denote by $\mathbf{e}$ the matrix of $\mathcal{M}_{n,1}(\mathbb{R})$ whose coefficients are all equal to 1, and $P = I_n - \frac{1}{n} \mathbf{e} \cdot \mathbf{e}^T$. We denote by $\Delta_n$ the set of EDM of order $n$ and $\Omega_n$ the set of symmetric positive matrices of order $n$ such that $M \cdot \mathbf{e} = 0$. We denote by $T$ the application from $\Delta_n$ to $\mathcal{M}_n(\mathbb{R})$ which associates to $D$ $$T(D) = -\frac{1}{2} P D P.$$
Let $D \in \Delta_n$. Let $A_1, \ldots, A_n$ be points whose matrix $D$ is the Euclidean distance matrix. We denote by $x_i$ the coordinate vectors of the $A_i$ and $M_A$ the matrix whose columns are the $x_i$ and $C$ the column formed by the $\|x_i\|^2$. Write $D$ as a linear combination of $C\mathbf{e}^T$, $\mathbf{e}C^T$ and $M_A^T \cdot M_A$. Deduce that for every matrix $D$ of $\Delta_n$ we have $T(D) \in \Omega_n$.

Let $H$ be a Hadamard matrix of order $n$ with first row constant equal to 1. Let $\lambda_1, \ldots, \lambda_n$ be real numbers such that $$\lambda_1 > 0 \geq \lambda_2 \geq \ldots \geq \lambda_n$$ and $$\sum_{i=1}^{n} \lambda_i = 0.$$ We denote by $U$ the matrix $\frac{1}{\sqrt{n}} H$ and $\Lambda$ the diagonal matrix whose diagonal coefficients are the $\lambda_i$. We finally denote by $D = U^T \Lambda U$.
Give a Euclidean distance matrix of order 4 such that its spectrum is $\{5, -1, -2, -2\}$.

Let $Z \in \mathscr{M}_{d}(\mathbb{R})$ be an invertible matrix with $\mathrm{S} = Z^{T}Z$, and let $(\lambda_{i})_{1 \leqslant i \leqslant d}$ be the decreasing family of strictly positive eigenvalues of $S$ with associated orthonormal basis $(u_{1}, \ldots, u_{d})$. We consider $v_{i} = \frac{1}{\sqrt{\lambda_{i}}} Z u_{i}$ for all $1 \leqslant i \leqslant d$.

• [(a)] Show that $(v_{1}, \ldots, v_{d})$ is an orthonormal basis of $\mathbb{R}^{d}$.
• [(b)] Verify that if $U = (u_{1} | \ldots | u_{d})$, $V = (v_{1} | \ldots | v_{d})$ and $D = \operatorname{Diag}(\sqrt{\lambda_{1}}, \ldots, \sqrt{\lambda_{d}})$ then $Z = VDU^{T}$.

Express in the form $Z = VDU^{T}$ (specifying your choices of $U$, $V$ and $D$) for the matrices $$Z_{1} = \begin{pmatrix} 0 & 1 \\ 2 & 0 \end{pmatrix} \quad \text{and} \quad Z_{2} = \begin{pmatrix} 0 & 1 \\ -2 & 0 \end{pmatrix}.$$

Give the value of $\delta(\boldsymbol{x}, \boldsymbol{y})$ as a function of $V_{n}(\boldsymbol{x})$, $V_{n}(\boldsymbol{y})$ and the singular values of $Z(\boldsymbol{x}, \boldsymbol{y})$ in the case where $\operatorname{det}(Z(\boldsymbol{x}, \boldsymbol{y})) > 0$.

Give the value of $\delta(\boldsymbol{x}, \boldsymbol{y})$ as a function of $V_{n}(\boldsymbol{x})$, $V_{n}(\boldsymbol{y})$ and the singular values of $Z(\boldsymbol{x}, \boldsymbol{y})$ in the case where $\operatorname{det}(Z(\boldsymbol{x}, \boldsymbol{y})) < 0$.

We denote by $D$ the diagonal matrix of size $n$: $$D = \operatorname{Diag}\left((1 - \alpha_j^2)_{1 \leq j \leq n}\right)$$ and $V \in \mathcal{M}_n(\mathbf{R})$ the matrix such that for every $j \in \llbracket 1, n \rrbracket$, the $j$-th column of $V$ is $V_j = f_j(S^\top) U$. Show that $$J(p) = V D V^\top.$$

We now consider the symmetric matrix $A$. By virtue of the spectral theorem, we denote by $\lambda_1 \leqslant \cdots \leqslant \lambda_n$ the eigenvalues of $A$, and $\left(\mathbf{w}_1, \ldots, \mathbf{w}_n\right)$ a corresponding orthonormal basis of eigenvectors.
(a) Show that $$A = \sum_{k=1}^n \lambda_k \mathbf{w}_k \mathbf{w}_k^T$$ (b) Show that for all $x \in \mathbb{R} \backslash \left\{\lambda_1, \ldots, \lambda_n\right\}$, we have $$\left(x \mathbb{I}_n - A\right)^{-1} = \sum_{k=1}^n \frac{1}{x - \lambda_k} \mathbf{w}_k \mathbf{w}_k^T$$

Let $\left(\mathbf{v}_1, \ldots, \mathbf{v}_n\right)$ be any orthonormal basis of $\mathbb{R}^n$. Show that $$\mathbb{I}_n = \sum_{k=1}^n \mathbf{v}_k \mathbf{v}_k^T.$$

We now focus on the symmetric matrix $A \in \mathcal{S}_n(\mathbb{R})$. By virtue of the spectral theorem, we denote by $\lambda_1 \leqslant \cdots \leqslant \lambda_n$ the eigenvalues of $A$, and $\left(\mathbf{w}_1, \ldots, \mathbf{w}_n\right)$ a corresponding orthonormal basis of eigenvectors.
(a) Show that $$A = \sum_{k=1}^n \lambda_k \mathbf{w}_k \mathbf{w}_k^T.$$
(b) Show that for all $x \in \mathbb{R} \backslash \left\{\lambda_1, \ldots, \lambda_n\right\}$, we have $$\left(x \mathbb{I}_n - A\right)^{-1} = \sum_{k=1}^n \frac{1}{x - \lambda_k} \mathbf{w}_k \mathbf{w}_k^T.$$