Show that $CL(\mathbf{R})$ is a vector space. Also show that $CL(\mathbf{R})$ is closed under multiplication.

Show that $CL(\mathbf{R})$ is a vector space. Also show that $CL(\mathbf{R})$ is stable under multiplication.

Show that if $H$ is a Hadamard matrix of order $n$ then there exists a Hadamard matrix of order $n$ whose coefficients of the first row are all equal to 1. Deduce that if $n \geq 2$ then $n$ is even.

Show that an adjacency matrix of a non-empty graph is never of rank 1.

Show that the following three assertions are equivalent
(i) $R_u = +\infty$,
(ii) $\mathbb{M}_n(u) = \mathscr{M}_n(\mathbb{C})$,
(iii) $\mathbb{M}_n(u) \neq \emptyset$ and $\forall A \in \mathbb{M}_n(u), \forall B \in \mathbb{M}_n(u), A + B \in \mathbb{M}_n(u)$, and give an example of a sequence $u$ satisfying these three assertions and such that $u_k \neq 0$ for every $k \in \mathbb{N}$.

Show that if $H$ is a Hadamard matrix of order $n$ greater than or equal to 4, then $n$ is a multiple of 4. One may begin by showing that we can assume the first row of $H$ is composed only of 1's and its second row is composed of $n/2$ coefficients equal to 1 then $n/2$ coefficients equal to $-1$.

Show that an adjacency matrix of a graph whose non-isolated vertices form a star-type graph is of rank 2 and represent an example of a graph whose adjacency matrix is of rank 2 and which is not of the previous type.

Let $D = \operatorname{Diag}(\alpha_{1}, \ldots, \alpha_{d})$ be a diagonal matrix with positive coefficients and let $R \in \mathrm{O}_{d}(\mathbb{R})$.

• [(a)] Show that for all $1 \leqslant i \leqslant d$, we have $|R_{ii}| \leqslant 1$ where $R_{ii}$ is the $i$-th diagonal coefficient of $R$.
• [(b)] Deduce that $\langle D, R \rangle \leqslant \operatorname{tr}(D)$.

Let $f$ be a symmetric endomorphism of $\mathbf{R}^n$. We denote by $\lambda_1 \leqslant \ldots \leqslant \lambda_n$ the eigenvalues ordered in increasing order of $f$.
Justify the existence of an orthonormal basis $(e_1, \ldots, e_n)$ of $\mathbf{R}^n$ formed of eigenvectors of $f$, the vector $e_i$ being associated with $\lambda_i$ for all $i \in \{1, \ldots, n\}$. We keep this basis henceforth.

Let $m \in \mathbb{N}$. We consider the matrix $$M = \left(\begin{array}{cccccc} \binom{0}{0} & 0 & \cdots & \cdots & \cdots & 0 \\ \binom{1}{0} & \binom{1}{1} & 0 & & & \vdots \\ \vdots & & \ddots & \ddots & & \vdots \\ \vdots & & & \ddots & \ddots & \vdots \\ \binom{m-1}{0} & & & & \binom{m-1}{m-1} & 0 \\ \binom{m}{0} & \cdots & \cdots & \cdots & \cdots & \binom{m}{m} \end{array}\right) \in \mathscr{M}_{m+1}(\mathbb{R}).$$ Justify that the families $\left(1, X, \ldots, X^{m}\right)$ and $\left(1, (X-1), \ldots, (X-1)^{m}\right)$ are bases of $\mathbb{R}_{m}[X]$.

Let $m \in \mathbb{N}$. We consider the matrix $$M = \left(\begin{array}{cccccc} \binom{0}{0} & 0 & \cdots & \cdots & \cdots & 0 \\ \binom{1}{0} & \binom{1}{1} & 0 & & & \vdots \\ \vdots & & \ddots & \ddots & & \vdots \\ \vdots & & & \ddots & \ddots & \vdots \\ \binom{m-1}{0} & & & & \binom{m-1}{m-1} & 0 \\ \binom{m}{0} & \cdots & \cdots & \cdots & \cdots & \binom{m}{m} \end{array}\right) \in \mathscr{M}_{m+1}(\mathbb{R}).$$ Show that the transpose of $M$ is the matrix of the linear map identity $$\begin{array}{ccc} \mathbb{R}_{m}[X] & \longrightarrow & \mathbb{R}_{m}[X] \\ P & \longmapsto & P \end{array}$$ in the bases $\left(1, X, \ldots, X^{m}\right)$ at the start and $\left(1, (X-1), \ldots, (X-1)^{m}\right)$ at the end.

Let $m \in \mathbb{N}$. We consider the matrix $$M = \left(\begin{array}{cccccc} \binom{0}{0} & 0 & \cdots & \cdots & \cdots & 0 \\ \binom{1}{0} & \binom{1}{1} & 0 & & & \vdots \\ \vdots & & \ddots & \ddots & & \vdots \\ \vdots & & & \ddots & \ddots & \vdots \\ \binom{m-1}{0} & & & & \binom{m-1}{m-1} & 0 \\ \binom{m}{0} & \cdots & \cdots & \cdots & \cdots & \binom{m}{m} \end{array}\right) \in \mathscr{M}_{m+1}(\mathbb{R}).$$ Establish that $M$ is invertible and explicitly determine its inverse.

Let $m \in \mathbb{N}$. We consider the matrix $$M = \left(\begin{array}{cccccc} \binom{0}{0} & 0 & \cdots & \cdots & \cdots & 0 \\ \binom{1}{0} & \binom{1}{1} & 0 & & & \vdots \\ \vdots & & \ddots & \ddots & & \vdots \\ \vdots & & & \ddots & \ddots & \vdots \\ \binom{m-1}{0} & & & & \binom{m-1}{m-1} & 0 \\ \binom{m}{0} & \cdots & \cdots & \cdots & \cdots & \binom{m}{m} \end{array}\right) \in \mathscr{M}_{m+1}(\mathbb{R}).$$ Deduce that for all $\left(u_{0}, \ldots, u_{m}\right), \left(v_{0}, \ldots, v_{m}\right) \in \mathbb{R}^{m+1}$, $$\text{if} \quad \forall k \leqslant m, \quad u_{k} = \sum_{\ell=0}^{k} \binom{k}{\ell} v_{\ell}, \quad \text{then} \quad \forall k \leqslant m, \quad v_{k} = \sum_{\ell=0}^{k} (-1)^{k-\ell} \binom{k}{\ell} u_{\ell}$$

Let $m \in \mathbb{N}$. We consider the matrix $$M = \left(\begin{array}{cccccc} \binom{0}{0} & 0 & \cdots & \cdots & \cdots & 0 \\ \binom{1}{0} & \binom{1}{1} & 0 & & & \vdots \\ \vdots & & \ddots & \ddots & & \vdots \\ \vdots & & & \ddots & \ddots & \vdots \\ \binom{m-1}{0} & & & & \binom{m-1}{m-1} & 0 \\ \binom{m}{0} & \cdots & \cdots & \cdots & \cdots & \binom{m}{m} \end{array}\right) \in \mathscr{M}_{m+1}(\mathbb{R}).$$ Justify that the families $\left(1, X, \ldots, X^m\right)$ and $\left(1, (X-1), \ldots, (X-1)^m\right)$ are bases of $\mathbb{R}_m[X]$.

Let $m \in \mathbb{N}$. We consider the matrix $$M = \left(\begin{array}{cccccc} \binom{0}{0} & 0 & \cdots & \cdots & \cdots & 0 \\ \binom{1}{0} & \binom{1}{1} & 0 & & & \vdots \\ \vdots & & \ddots & \ddots & & \vdots \\ \vdots & & & \ddots & \ddots & \vdots \\ \binom{m-1}{0} & & & & \binom{m-1}{m-1} & 0 \\ \binom{m}{0} & \cdots & \cdots & \cdots & \cdots & \binom{m}{m} \end{array}\right) \in \mathscr{M}_{m+1}(\mathbb{R}).$$ Show that the transpose of $M$ is the matrix of the linear map identity $$\begin{array}{ccc} \mathbb{R}_m[X] & \longrightarrow & \mathbb{R}_m[X] \\ P & \longmapsto & P \end{array}$$ in the bases $\left(1, X, \ldots, X^m\right)$ at the start and $\left(1, (X-1), \ldots, (X-1)^m\right)$ at the end.

Let $m \in \mathbb{N}$. We consider the matrix $$M = \left(\begin{array}{cccccc} \binom{0}{0} & 0 & \cdots & \cdots & \cdots & 0 \\ \binom{1}{0} & \binom{1}{1} & 0 & & & \vdots \\ \vdots & & \ddots & \ddots & & \vdots \\ \vdots & & & \ddots & \ddots & \vdots \\ \binom{m-1}{0} & & & & \binom{m-1}{m-1} & 0 \\ \binom{m}{0} & \cdots & \cdots & \cdots & \cdots & \binom{m}{m} \end{array}\right) \in \mathscr{M}_{m+1}(\mathbb{R}).$$ Establish that $M$ is invertible and explicitly determine its inverse.

Let $m \in \mathbb{N}$. We consider the matrix $$M = \left(\begin{array}{cccccc} \binom{0}{0} & 0 & \cdots & \cdots & \cdots & 0 \\ \binom{1}{0} & \binom{1}{1} & 0 & & & \vdots \\ \vdots & & \ddots & \ddots & & \vdots \\ \vdots & & & \ddots & \ddots & \vdots \\ \binom{m-1}{0} & & & & \binom{m-1}{m-1} & 0 \\ \binom{m}{0} & \cdots & \cdots & \cdots & \cdots & \binom{m}{m} \end{array}\right) \in \mathscr{M}_{m+1}(\mathbb{R}).$$ Deduce that for all $\left(u_0, \ldots, u_m\right), \left(v_0, \ldots, v_m\right) \in \mathbb{R}^{m+1}$, $$\text{if} \quad \forall k \leqslant m, \quad u_k = \sum_{\ell=0}^{k} \binom{k}{\ell} v_\ell, \quad \text{then} \quad \forall k \leqslant m, \quad v_k = \sum_{\ell=0}^{k} (-1)^{k-\ell} \binom{k}{\ell} u_\ell.$$

Let $f$ be a symmetric endomorphism of $\mathbf{R}^n$ with eigenvalues $\lambda_1 \leqslant \ldots \leqslant \lambda_n$ and associated orthonormal eigenbasis $(e_1, \ldots, e_n)$.
Let $k \in \llbracket 1, n \rrbracket$ and $S_k$ a vector subspace of $\mathbf{R}^n$ of dimension $k$. We set $T_k = \operatorname{Vect}(e_k, \ldots, e_n)$.
Justify that $S_k \cap T_k \neq \{0\}$.

We assume in this question that $0 < R_u \leqslant 1$. Let $A \in \mathbb{M}_n(u)$ and $B \in \mathbb{M}_n(u)$ be two symmetric matrices such that $AB = BA$. Show that $AB \in \mathbb{M}_n(u)$.

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 $M = \lambda u \cdot u^T - B^T \cdot B$.

Determine the characteristic polynomial of the double star with $d _ { 1 } + d _ { 2 } + 2$ vertices, consisting respectively of two disjoint stars with $d _ { 1 }$ and $d _ { 2 }$ branches, to which an additional edge has been added connecting the two centers of the two stars. What is the rank of the adjacency matrix of this double star?

We denote by $\mathbf{e}$ the matrix of $\mathcal{M}_{n,1}(\mathbb{R})$ whose coefficients are all equal to 1, and $P$ the matrix of order $n$ defined by $$P = I_n - \frac{1}{n} \mathbf{e} \cdot \mathbf{e}^T.$$
Show that $P$ is symmetric and that the endomorphism of $\mathbf{R}^n$ canonically associated is an orthogonal projection onto $\operatorname{Vect}(\mathbf{e})^\perp$.

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 $A \in \mathscr{M}_n(\mathbb{C})$. We denote by $\mathbb{C}[A]$ the set of elements of $\mathscr{M}_n(\mathbb{C})$ of the form $P(A)$ where $P \in \mathbb{C}[X]$ is a polynomial. We denote $$(\mathbb{C}[A])^* = \left\{B \in \mathbb{C}[A] \cap \mathrm{GL}_n(\mathbb{C}) \mid B^{-1} \in \mathbb{C}[A]\right\}$$ Show that there exists an open set $U$ of $\mathbb{C}[A]$ containing $0$ and an open set $V$ of $\mathbb{C}[A]$ containing the identity matrix $I_n$ such that the exponential function induces a continuous bijection from $U \subset \mathbb{C}[A]$ to $V$ whose inverse is a continuous function on $V$.