Give two definitions of a vector isometry of $\mathbb{R}^n$ and prove their equivalence.

Prove that if $M \in \mathrm{O}_n(\mathbb{R})$, then its determinant equals 1 or $-1$. What do you think of the converse?

Prove that $\mathcal{P}_n = \mathcal{X}_n \cap \mathrm{O}_n(\mathbb{R})$ and determine its cardinality.

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, defined on a probability space $(\Omega, \mathcal{A}, P)$ and following the same Bernoulli distribution with parameter $p$. Let $S = X_1 + \ldots + X_n$.
If $\omega \in \Omega$, we introduce the column matrix $$U(\omega) = \begin{pmatrix} X_1(\omega) \\ \vdots \\ X_n(\omega) \end{pmatrix}$$ and the matrix $M(\omega) = U(\omega)\, {}^t(U(\omega))$. The application $M : \left\{\begin{array}{l} \Omega \rightarrow \mathcal{M}_n(\mathbb{R}) \\ \omega \mapsto M(\omega) \end{array}\right.$ is thus a random variable.
a) If $\omega \in \Omega$, justify that $M(\omega) \in \mathcal{X}_n$.
b) If $\omega \in \Omega$, justify that $\operatorname{tr}(M(\omega)) \in \{0, \ldots, n\}$, that $M(\omega)$ is diagonalizable over $\mathbb{R}$ and that $\operatorname{rg}(M(\omega)) \leqslant 1$.
c) If $\omega \in \Omega$, justify that $M(\omega)$ is an orthogonal projection matrix if and only if $S(\omega) \in \{0,1\}$.

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 $S = X_1 + \ldots + X_n$, $U(\omega) = (X_1(\omega), \ldots, X_n(\omega))^T$ and $M(\omega) = U(\omega)\,{}^t(U(\omega))$.
Express $M^k$ in terms of $S$ and $M$.
What is the probability that the sequence of matrices $(M^k)_{k \in \mathbb{N}}$ is convergent?
Show that, in this case, the limit is a projection matrix.

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

For the positive matrix $A$ of $\mathcal{M}_n(\mathbb{R})$, show that the following conditions are equivalent:

• the matrix $A$ is irreducible;
• the matrix $B = I_n + A + A^2 + \cdots + A^{n-1}$ is strictly positive;
• the matrix $C = (I_n + A)^{n-1}$ is strictly positive.

Let $A$ be irreducible. Show that no row (and no column) of $A$ is identically zero.

In this question, $A$ is a given irreducible matrix.
Suppose that $\forall i \in \llbracket 1, n \rrbracket, a_{i,i} > 0$. Show that $A^{n-1} > 0$ (so $A$ is primitive). Reason in terms of paths in $A$.

In this question, $A$ is a given irreducible matrix.
Suppose that: $\exists i \in \llbracket 1, n \rrbracket, a_{i,i} > 0$. Show that $A$ is primitive.
For all $j$ and $k$ in $\llbracket 1, n \rrbracket$, one can show that there exists in $A$ a path from $j$ to $k$ passing through $i$, and consider the maximum $m$ of the lengths of the paths thus obtained. One will prove that $A^m > 0$.

Let $A$ be an imprimitiv matrix with coefficient of imprimitivity $p \geqslant 2$.
For any integer $m$ that is not a multiple of $p$, show that the diagonal of $A^m$ is identically zero. One can be interested in the trace of $A^m$.
Deduce that the result of question IV.B.3 no longer holds if $A$ is imprimitiv.

We define the matrix $Z_n = (z_{i,j}) \in \mathcal{M}_n(\mathbb{R})$ by $z_{i,j} = \begin{cases} 1 & \text{if } 1 \leqslant i < n \text{ and } j = i+1 \\ 1 & \text{if } (i,j) \in \{(n-1,1),(n,2)\} \\ 0 & \text{in all other cases} \end{cases}$
Show that the matrix $Z_n$ is irreducible.

We define the matrix $Z_n = (z_{i,j}) \in \mathcal{M}_n(\mathbb{R})$ by $z_{i,j} = \begin{cases} 1 & \text{if } 1 \leqslant i < n \text{ and } j = i+1 \\ 1 & \text{if } (i,j) \in \{(n-1,1),(n,2)\} \\ 0 & \text{in all other cases} \end{cases}$
Show that the characteristic polynomial of $Z_n$ is $X(X^{n-1} - 2)$.
Deduce that $Z_n$ is imprimitiv and specify its coefficient of imprimitivity.

We define the matrix $Z_n = (z_{i,j}) \in \mathcal{M}_n(\mathbb{R})$ by $z_{i,j} = \begin{cases} 1 & \text{if } 1 \leqslant i < n \text{ and } j = i+1 \\ 1 & \text{if } (i,j) \in \{(n-1,1),(n,2)\} \\ 0 & \text{in all other cases} \end{cases}$
Show that $Z_n^{n^2-2n+2} = 2^{n-1} Z_n$ and recover the fact that $Z_n$ is not primitive.

Let $A \geqslant 0$ in $\mathcal{M}_n(\mathbb{R})$, an irreducible matrix. We denote $r$ its spectral radius. Let $p \geqslant 1$ be the coefficient of imprimitivity of $A$ (reminder: by convention, $p = 1$ if $A$ is primitive). Let $\chi_A(X) = X^n + c_{k_1}X^{n-k_1} + c_{k_1}X^{n-k_2} + \cdots + c_{k_s}X^{n-k_s}$ be its characteristic polynomial, written according to decreasing powers and showing only the nonzero coefficients $c_k$.
We recall that the spectrum of $A$ is invariant under the map $z \mapsto \omega z$, where $\omega = \exp(2\mathrm{i}\pi/p)$.
Deduce that, for all $k \in \{k_1, k_2, \ldots, k_s\}$, the integer $k$ is divisible by $p$. Think of the elementary symmetric functions of the $\lambda_i$.

Let $A \geqslant 0$ in $\mathcal{M}_n(\mathbb{R})$, an irreducible matrix. We denote $r$ its spectral radius. Let $p \geqslant 1$ be the coefficient of imprimitivity of $A$. Let $\chi_A(X) = X^n + c_{k_1}X^{n-k_1} + c_{k_1}X^{n-k_2} + \cdots + c_{k_s}X^{n-k_s}$ be its characteristic polynomial, written according to decreasing powers and showing only the nonzero coefficients $c_k$. We will show that $p$ is the gcd of the integers $k_1, k_2, \ldots, k_s$.
Conversely, we assume by contradiction that the $k_j$ are all divisible by $qp$, with $q \geqslant 2$. We set $\beta = \mathrm{e}^{2\mathrm{i}\pi/(qp)}$ (so $\beta^q = \omega$). Show that $\beta r$ is an eigenvalue of $A$ and conclude.

Let $A \in \mathcal{M}_n(\mathbb{R})$ be an irreducible matrix. For all $i$ in $\llbracket 1, n \rrbracket$, we denote $L_i = \{m \in \mathbb{N}^*, a_{i,i}^{(m)} > 0\}$ the (nonempty) set of lengths of circuits of $A$ passing through $i$, and we denote $d_i$ the gcd of the elements of $L_i$.
Establish that the coefficient of imprimitivity $p$ of $A$ is equal to $d_i$ for all $i$ in $\llbracket 1, n \rrbracket$ (this gcd does not depend on the index $i$).

Give the matrix $M = \left(M_{i,j}\right)_{1 \leqslant i,j \leqslant n+1}$ of $\tau$ in the basis $\left(P_k\right)_{k \in \llbracket 1, n+1 \rrbracket}$. Express the coefficients $M_{i,j}$ in terms of $i$ and $j$.

Is the application $\tau$ bijective? If so, specify $\tau^{-1}$. Is the expression of $\tau^j$ found in question I.A.2 for $j \in \mathbb{N}$ valid for $j \in \mathbb{Z}$?

What is $M^{-1}$? Express the coefficients $\left(M^{-1}\right)_{i,j}$ in terms of $i$ and $j$.

We are given a real sequence $\left(u_k\right)_{k \in \mathbb{N}}$ and we define for every integer $k \in \mathbb{N}$ $$v_k = \sum_{j=0}^{k} \binom{k}{j} u_j$$ Determine a matrix $Q \in \mathcal{M}_{n+1}(\mathbb{R})$ such that $$\left(\begin{array}{c} v_0 \\ v_1 \\ \vdots \\ v_n \end{array}\right) = Q \left(\begin{array}{c} u_0 \\ u_1 \\ \vdots \\ u_n \end{array}\right)$$