Let $\ell$ and $m$ be two $n$-tuples of real numbers. We write $$\ell \preccurlyeq m \quad \text{if and only if, for every integer } j,\, 1 \leqslant j \leqslant n, \quad \ell_{j} \leqslant m_{j}.$$ Let $L, M \in \mathcal{S}_{n}(\mathbb{R})$ such that $(0, \ldots, 0) \preccurlyeq s^{\downarrow}(M - L)$. Show that $s^{\downarrow}(L) \preccurlyeq s^{\downarrow}(M)$.

Let $\ell$ and $m$ be two $n$-tuples of real numbers. We write $$\ell \preccurlyeq m \quad \text{if and only if, for every integer } j,\, 1 \leqslant j \leqslant n, \quad \ell_{j} \leqslant m_{j}.$$ Show that for every matrix $M \in \mathcal{S}_{n}(\mathbb{R})$, $(0, \ldots, 0) \preccurlyeq s^{\downarrow}\left(\|M\| I_{n} - M\right)$.

Let $\ell$ and $m$ be two $n$-tuples of real numbers. We write $$\ell \preccurlyeq m \quad \text{if and only if, for every integer } j,\, 1 \leqslant j \leqslant n, \quad \ell_{j} \leqslant m_{j}.$$ Let $L, M \in \mathcal{S}_{n}(\mathbb{R})$, we denote by $m = s^{\downarrow}(M)$ and $\ell = s^{\downarrow}(L)$. Show that $$\max_{1 \leqslant j \leqslant n} \left|\ell_{j} - m_{j}\right| \leqslant \|L - M\|.$$

Conclude that the function $s^{\downarrow} : \mathcal{S}_{n}(\mathbb{R}) \rightarrow \mathbb{R}^{n}$ is continuous.

We denote by $\mathcal{S}_{n}^{\dagger}(\mathbb{R})$ the set of $n \times n$ symmetric matrices whose eigenvalues are all simple. Show that the first component $s_{1}^{\downarrow}$ of $s^{\downarrow}$ is of class $\mathscr{C}^{1}$ on $\mathcal{S}_{2}^{\dagger}(\mathbb{R})$, but not on $\mathcal{S}_{2}(\mathbb{R})$. (One may use question 1d.)

In this part, we consider two real symmetric matrices $A, B \in \mathcal{S}_{n}(\mathbb{R})$ and their sum $C = A + B$. We denote by $a = s^{\downarrow}(A)$, $b = s^{\downarrow}(B)$ and $c = s^{\downarrow}(C)$.
Show that $$\sum_{i=1}^{n} c_{i} = \sum_{i=1}^{n} a_{i} + \sum_{i=1}^{n} b_{i}.$$

In this part, we consider two real symmetric matrices $A, B \in \mathcal{S}_{n}(\mathbb{R})$ and their sum $C = A + B$. We denote by $a = s^{\downarrow}(A)$, $b = s^{\downarrow}(B)$ and $c = s^{\downarrow}(C)$.
Show that $a_{1} + b_{1} \geqslant c_{1}$.

In this part, we consider two real symmetric matrices $A, B \in \mathcal{S}_{n}(\mathbb{R})$ and their sum $C = A + B$. We denote by $a = s^{\downarrow}(A)$, $b = s^{\downarrow}(B)$ and $c = s^{\downarrow}(C)$.
Show that $a_{n} + b_{n} \leqslant c_{n}$.

In this part, we consider two real symmetric matrices $A, B \in \mathcal{S}_{n}(\mathbb{R})$ and their sum $C = A + B$. We denote by $a = s^{\downarrow}(A)$, $b = s^{\downarrow}(B)$ and $c = s^{\downarrow}(C)$.
By using spectral resolutions of $A$, $B$ and $C$, show that if the strictly positive integers $j$ and $k$ satisfy $j + k \leqslant n + 1$, we have $$c_{j+k-1} \leqslant a_{j} + b_{k}.$$ Deduce that for every integer $j$, $1 \leqslant j \leqslant n$, $$a_{j} + b_{n} \leqslant c_{j}.$$

We denote by $a_{ii}$ for $1 \leqslant i \leqslant n$ the diagonal elements of $A \in \mathcal{S}_{n}(\mathbb{R})$ with ordered spectrum $a = s^{\downarrow}(A)$. Prove that $a_{11} \leqslant a_{1}$.

We denote by $a_{ii}$ for $1 \leqslant i \leqslant n$ the diagonal elements of $A \in \mathcal{S}_{n}(\mathbb{R})$ with ordered spectrum $a = s^{\downarrow}(A)$. Show that, more generally than in 8a, we have for every integer $1 \leqslant j \leqslant n$ $$\sum_{i=1}^{j} a_{ii} \leqslant \sum_{i=1}^{j} a_{i}.$$

In this part, we consider two real symmetric matrices $A, B \in \mathcal{S}_{n}(\mathbb{R})$ and their sum $C = A + B$. We denote by $a = s^{\downarrow}(A)$, $b = s^{\downarrow}(B)$ and $c = s^{\downarrow}(C)$. We denote by $\mathcal{R}_{j}$ the set of orthonormal families of cardinality $j$ in $\mathbb{R}^{n}$.
Conclude that for every integer $1 \leqslant j \leqslant n$ $$\sum_{i=1}^{j} c_{i} \leqslant \sum_{i=1}^{j} a_{i} + \sum_{i=1}^{j} b_{i}.$$

Prove that all elements of $\mathcal{P}_n$ are diagonalizable over $\mathbb{C}$.

Determine the eigenvectors common to all elements of $\mathcal{P}_n$ in the cases $n = 2$ and $n = 3$.

We propose to prove that the only vector subspaces of $\mathbb{R}^n$ stable under all $u_\sigma$, $\sigma \in S_n$ are $\{0_{\mathbb{R}^n}\}$, $\mathbb{R}^n$, the line $D$ generated by $e_1 + e_2 + \cdots + e_n$ and the hyperplane $H$ orthogonal to $D$.
a) Verify that these four vector subspaces are stable under all $u_\sigma$.
b) Let $V$ be a vector subspace of $\mathbb{R}^n$, not contained in $D$ and stable under all $u_\sigma$. Prove that there exists a pair $(i,j) \in \{1,\ldots,n\}^2$ with $i \neq j$ such that $e_i - e_j \in V$, then that the $n-1$ vectors $e_k - e_j$ ($k \in \{1,\ldots,n\}$, $k \neq j$) belong to $V$.
c) Conclude.

We are given a matrix $M$ of $\mathrm{GL}_n(\mathbb{R})$ whose coefficients are all natural integers and such that the set formed by all coefficients of all successive powers of $M$ is finite.
Prove that $M^{-1}$ has coefficients in $\mathbb{N}$ and deduce that $M$ is a permutation matrix. What can be said of the converse?

Let $p \in ]0,1[$. Let $X_1, \ldots, X_n$ be mutually independent random variables following the same Bernoulli distribution with parameter $p$. Let $U(\omega) = (X_1(\omega), \ldots, X_n(\omega))^T$ and $M(\omega) = U(\omega)\,{}^t(U(\omega))$.
What is the probability that $M$ has two distinct eigenvalues?

For any primitive matrix $A$ in $\mathcal{M}_n(\mathbb{R})$, we denote $r$ the spectral radius of $A$, $L = xy^\top$ where $x > 0$ is a direction vector of $\operatorname{Ker}(A - rI_n)$ and $y > 0$ is a direction vector of $\operatorname{Ker}(A^\top - rI_n)$ with $y^\top x = 1$. We set $B = A - rL$.
Let $\lambda$ be a nonzero eigenvalue of $B$ and let $z$ be an associated eigenvector. By contradiction, we assume $\rho(B) = r$. We can therefore choose $\lambda$ such that $|\lambda| = r$. Show that then $\lambda = r$ then $Lz = z$ and reach a contradiction. Conclude.

For any primitive matrix $A$ in $\mathcal{M}_n(\mathbb{R})$, we denote $r$ the spectral radius of $A$, $L = xy^\top$ where $x > 0$ is a direction vector of $\operatorname{Ker}(A - rI_n)$ and $y > 0$ is a direction vector of $\operatorname{Ker}(A^\top - rI_n)$ with $y^\top x = 1$. We set $B = A - rL$, and we have shown $\rho(B) < r$ and $\forall m \in \mathbb{N}^*, (A - rL)^m = A^m - r^m L$.
Deduce from the above (and from subsection IV.A) that $\lim_{m \rightarrow +\infty} \left(\frac{1}{r}A\right)^m = L$.

For any primitive matrix $A$ in $\mathcal{M}_n(\mathbb{R})$, we denote $r$ the spectral radius of $A$. Let $\mu$ be the multiplicity of $r$ as an eigenvalue of $A$ and let $T = PAP^{-1}$ be a triangular reduction of $A$.
By examining the diagonal of $\left(\frac{1}{r}T\right)^m$ when $m \rightarrow +\infty$, show that $\mu = 1$.

Let $A = (a_{i,j})$ in $\mathcal{M}_n(\mathbb{R})$, with $A \geqslant 0$. We say that $A$ is irreducible if, for all $i$ and $j$ in $\llbracket 1, n \rrbracket$, there exists $m \geqslant 0$ (depending a priori on $i$ and $j$) such that $a_{i,j}^{(m)} > 0$.
Express the irreducibility of $A$ in terms of paths in $A$.

Let $A = (a_{i,j})$ in $\mathcal{M}_n(\mathbb{R})$, with $A \geqslant 0$, irreducible.
Show that if $A$ is irreducible, then for all $i$ and $j$ in $\llbracket 1, n \rrbracket$, there exists $m \in \llbracket 0, n-1 \rrbracket$ (depending a priori on $i$ and $j$) such that $a_{i,j}^{(m)} > 0$.

Give a simple example of a square irreducible matrix that is not primitive.

Let $A = (a_{i,j})$ in $\mathcal{M}_n(\mathbb{R})$, with $A \geqslant 0$.
Show that if $A$ is not irreducible, then $A^2$ is not irreducible.
On the other hand, give a simple example of an irreducible matrix $A$ such that $A^2$ is not irreducible.

Show that the spectral radius of an irreducible matrix is strictly positive.