We consider $M \in \mathscr { D } _ { \rho } \left( S _ { n } ( \mathbb { R } ) \right)$. We fix a real eigenvalue $\lambda$ of $M _ { \mid t = 0 }$ and denote by $d$ its multiplicity as a root of $\chi _ { \mid t = 0 }$ where $\chi = \det(XI_n - M)$. We set $Q = ( B U \mid A V ) \in \mathscr { D } _ { \rho _ { 2 } } \left( \mathscr { M } _ { n } ( \mathbb { R } ) \right)$ and $Q^{-1} \cdot M \cdot Q = \operatorname{Diag}(M_1, M_2)$.
Show that there exists $\rho _ { 3 } \in \mathbb { R } _ { + } ^ { * }$ such that $\rho _ { 3 } \leqslant \rho _ { 2 }$ and matrices $R _ { 1 } \in \mathrm { GL } _ { d } \left( \mathscr { D } _ { \rho _ { 3 } } ( \mathbb { R } ) \right) , R _ { 2 } \in \mathrm { GL } _ { n - d } \left( \mathscr { D } _ { \rho _ { 3 } } ( \mathbb { R } ) \right)$ such that the matrix $Q \cdot \operatorname { Diag } \left( R _ { 1 } , R _ { 2 } \right)$ is orthogonal. (One may use the result of question 17.)

Prove Theorem 2: Let $M \in \mathscr { D } _ { \rho } \left( S _ { n } ( \mathbb { R } ) \right)$. Then there exists $r \in \mathbb { R } _ { + } ^ { * }$ such that $r \leqslant \rho$ and an orthogonal matrix $P \in \mathscr { D } _ { r } \left( \mathscr { M } _ { n } ( \mathbb { R } ) \right)$ such that $P ^ { \mathrm { T } } \cdot M \cdot P$ is diagonal.

Show that the matrices $M$ and $\left( m _ { \rho ( i ) , \rho ( j ) } \right) _ { 1 \leq i , j \leq n }$ are similar. Deduce that if $G = ( S , A )$ is a non-empty graph, and if $\sigma$ and $\sigma ^ { \prime }$ are two indexings of $S$, then $M _ { G , \sigma }$ and $M _ { G , \sigma ^ { \prime } }$ are similar.

Let $n$ be a natural integer with $n \geqslant 2$. For any real number $x$, we consider the following matrix in $\mathscr{M}_{n}(\mathbb{R})$ $$M_{x} = \left(\begin{array}{ccccc} x & 1 & \cdots & 1 & 1 \\ 1 & x & \cdots & 1 & 1 \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ 1 & 1 & \cdots & x & 1 \\ 1 & 1 & \cdots & 1 & x \end{array}\right)$$ Show that the matrix $-M_{0}$ is diagonalizable and determine its eigenvalues and eigenspaces.

Let $n$ be a natural integer with $n \geqslant 2$. For any real number $x$, we consider the following matrix in $\mathscr{M}_{n}(\mathbb{R})$ $$M_{x} = \left(\begin{array}{ccccc} x & 1 & \cdots & 1 & 1 \\ 1 & x & \cdots & 1 & 1 \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ 1 & 1 & \cdots & x & 1 \\ 1 & 1 & \cdots & 1 & x \end{array}\right)$$ Deduce that for all $x \in \mathbb{R}$, we have $$\sum_{\sigma \in \mathfrak{S}_{n}} \varepsilon(\sigma) x^{\nu(\sigma)} = (x-1)^{n-1}(x+n-1)$$

Let $n$ be a natural integer with $n \geqslant 2$. For any real number $x$, we consider the following matrix in $\mathscr{M}_n(\mathbb{R})$ $$M_x = \left(\begin{array}{ccccc} x & 1 & \cdots & 1 & 1 \\ 1 & x & \cdots & 1 & 1 \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ 1 & 1 & \cdots & x & 1 \\ 1 & 1 & \cdots & 1 & x \end{array}\right).$$ Show that the matrix $-M_0$ is diagonalizable and determine its eigenvalues and eigenspaces.

Justify that an adjacency matrix of a non-empty graph is diagonalizable.

Show that an adjacency matrix of a non-empty graph is never of rank 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 $f$ 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 $f$ 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 $S_k \cap T_k \neq \{0\}$.

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 $k \in \llbracket 1, n \rrbracket$, $S_k$ a vector subspace of $\mathbf{R}^n$ of dimension $k$, and $T_k = \operatorname{Vect}(e_k, \ldots, e_n)$.
By considering $x \in S_k \cap T_k$, justify that: $$\max_{x \in S_k, \|x\|=1} (x, f(x)) \geq \lambda_k.$$

Let $f$ be a symmetric endomorphism of $\mathbf{R}^n$ with eigenvalues $\lambda_1 \leqslant \ldots \leqslant \lambda_n$ and associated orthonormal eigenbasis $(e_1, \ldots, e_n)$. For $k \in \llbracket 1, n \rrbracket$, let $\pi_k$ denote the set of vector subspaces of $\mathbf{R}^n$ of dimension $k$.
Let $k \in \llbracket 1, n \rrbracket$. Using $S = \operatorname{Vect}(e_1, \ldots, e_k) \in \pi_k$, show the equality: $$\lambda_k = \min_{S \in \pi_k} \left( \max_{x \in S, \|x\|=1} (x, f(x)) \right)$$ This is the Courant-Fischer theorem.

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 $P$ 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. We denote by $\Omega_n$ the set of symmetric positive matrices of order $n$ such that $M \cdot \mathbf{e} = 0$. We denote by $K$ the application from $\Omega_n$ to $\mathcal{M}_n(\mathbb{R})$ which associates to a matrix $A$ $$K(A) = \mathbf{e} \cdot \mathbf{a}^T + \mathbf{a} \cdot \mathbf{e}^T - 2A$$ where $\mathbf{a}$ is the column matrix of $\mathcal{M}_{n,1}(\mathbb{R})$ whose coefficients are the diagonal coefficients of $A$.
Show that for every matrix $A$ of $\Omega_n$ we have $K(A) \in \Delta_n$.

We denote by $\mathbf{e}$ the matrix of $\mathcal{M}_{n,1}(\mathbb{R})$ whose coefficients are all equal to 1, $P = I_n - \frac{1}{n}\mathbf{e}\cdot\mathbf{e}^T$, $\Delta_n$ the set of EDM of order $n$, and $\Omega_n$ the set of symmetric positive matrices of order $n$ such that $M \cdot \mathbf{e} = 0$. The application $T: \Delta_n \to \mathcal{M}_n(\mathbb{R})$ associates to $D$ the matrix $T(D) = -\frac{1}{2}PDP$, and the application $K: \Omega_n \to \mathcal{M}_n(\mathbb{R})$ associates to $A$ the matrix $K(A) = \mathbf{e}\cdot\mathbf{a}^T + \mathbf{a}\cdot\mathbf{e}^T - 2A$ where $\mathbf{a}$ is the column of diagonal coefficients of $A$.
Show that the applications $T: \Delta_n \rightarrow \Omega_n$ and $K: \Omega_n \rightarrow \Delta_n$ satisfy: $$T \circ K = \operatorname{Id}_{\Omega_n}.$$

We denote by $\mathbf{e}$ the matrix of $\mathcal{M}_{n,1}(\mathbb{R})$ whose coefficients are all equal to 1, $P = I_n - \frac{1}{n}\mathbf{e}\cdot\mathbf{e}^T$, and $\Delta_n$ the set of EDM of order $n$.
Show that a symmetric matrix $D$ of order $n$ with non-negative coefficients and zero diagonal is EDM if and only if $-\frac{1}{2}PDP$ is positive.

We denote by $\mathbf{e}$ the matrix of $\mathcal{M}_{n,1}(\mathbb{R})$ whose coefficients are all equal to 1, $P = I_n - \frac{1}{n}\mathbf{e}\cdot\mathbf{e}^T$, and $\Delta_n$ the set of EDM of order $n$.
Show that every non-zero symmetric matrix with non-negative coefficients and zero diagonal, having a unique strictly positive eigenvalue with eigenspace of dimension 1 and eigenvector $\mathbf{e}$, is EDM.

We denote by $\mathbf{e}$ the matrix of $\mathcal{M}_{n,1}(\mathbb{R})$ whose coefficients are all equal to 1, $P = I_n - \frac{1}{n}\mathbf{e}\cdot\mathbf{e}^T$, and $\Delta_n$ the set of EDM of order $n$.
Specify the sum $\displaystyle\sum_{i=1}^{n} \lambda_i$ of the eigenvalues of an EDM of order $n$.

Let $Z \in \mathscr{M}_{d}(\mathbb{R})$ be an invertible matrix. We denote $\mathrm{S} = Z^{T}Z$. Show that there exists a decreasing family $(\lambda_{i})_{1 \leqslant i \leqslant d}$ of strictly positive real numbers and an orthonormal basis $(u_{1}, \ldots, u_{d})$ of $\mathbb{R}^{d}$ such that $Su_{i} = \lambda_{i} u_{i}$ for all $1 \leqslant i \leqslant d$.

We denote by $\mathbf{e}$ the matrix of $\mathcal{M}_{n,1}(\mathbb{R})$ whose coefficients are all equal to 1, $P = I_n - \frac{1}{n}\mathbf{e}\cdot\mathbf{e}^T$, and $\Delta_n$ the set of EDM of order $n$.
Let $D$ be a non-zero EDM of order $n$. Show that for all $x \in \operatorname{Vect}(\mathbf{e})^\perp$, we have $$x^T D x \leqslant 0.$$

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 $Z = VDU^{T}$.

We denote by $\mathbf{e}$ the matrix of $\mathcal{M}_{n,1}(\mathbb{R})$ whose coefficients are all equal to 1, $P = I_n - \frac{1}{n}\mathbf{e}\cdot\mathbf{e}^T$, and $\Delta_n$ the set of EDM of order $n$.
Let $D$ be a non-zero EDM of order $n$. Let $\lambda_1, \ldots, \lambda_n$ be its eigenvalues, ordered in increasing order. Show $$\lambda_{n-1} \leqslant 0$$ and deduce that $D$ has exactly one strictly positive eigenvalue.

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