Let $M \in \mathcal{Y}_n$ and $\lambda$ a complex eigenvalue of $M$. Prove that $|\lambda| \leqslant n$ and give an explicit example where equality holds.
List the elements of $\mathcal{X}_2' = \mathcal{X}_2 \cap \mathrm{GL}_2(\mathbb{R})$. Specify (by justifying) which ones are diagonalizable over $\mathbb{R}$.
Prove that $\mathcal{X}_2' = \mathcal{X}_2 \cap \mathrm{GL}_2(\mathbb{R})$ generates the vector space $\mathcal{M}_2$. For $n \geqslant 2$, does $\mathcal{X}_n'$ generate the vector space $\mathcal{M}_n(\mathbb{R})$?
For all $(M, N) \in (\mathcal{M}_n(\mathbb{R}))^2$, we denote $$(M \mid N) = \operatorname{tr}({}^t M N)$$ Prove that this defines an inner product on $\mathcal{M}_n(\mathbb{R})$. Explicitly express $(M \mid N)$ in terms of the coefficients of $M$ and $N$.
We denote by $\|M\|$ the Euclidean norm associated with the inner product $(M \mid N) = \operatorname{tr}({}^t M N)$. Fix $A \in \mathcal{M}_n(\mathbb{R})$, prove that there exists a matrix $M \in \mathcal{Y}_n$ such that: $$\forall N \in \mathcal{Y}_n \quad \|A - M\| \leqslant \|A - N\|$$
We denote by $\|M\|$ the Euclidean norm associated with the inner product $(M \mid N) = \operatorname{tr}({}^t M N)$. Justify the uniqueness of the matrix $M \in \mathcal{Y}_n$ minimizing $\|A - M\|$ over $\mathcal{Y}_n$ and explicitly express its coefficients in terms of those of $A$.
Let $J \in \mathcal{X}_n$ be the matrix whose coefficients all equal 1. We set $M = J - I_n$. Calculate $\operatorname{det}(M)$ and deduce that $\lim_{k \to +\infty} y_k = +\infty$.
Let $N = (n_{i,j})_{i,j} \in \mathcal{Y}_n$. Fix $1 \leqslant i, j \leqslant n$ and suppose that $n_{i,j} \in ]0,1[$. Prove that by replacing $n_{i,j}$ either by 0 or by 1, we can obtain a matrix $N'$ of $\mathcal{Y}_n$ such that $\operatorname{det}(N) \leqslant \operatorname{det}(N')$. Deduce that $x_n = y_n$.
Let $\sigma$ and $\sigma'$ be two elements of $S_n$. Prove that $P_\sigma P_{\sigma'} = P_{\sigma \circ \sigma'}$. Justify that the application $\left\{\begin{array}{l} \mathbb{Z} \rightarrow S_n \\ k \mapsto \sigma^k \end{array}\right.$ is not injective. Deduce that there exists an integer $N \geqslant 1$ such that $\sigma^N = \operatorname{Id}_{\{1,\ldots,n\}}$, where $\operatorname{Id}_{\{1,\ldots,n\}}$ denotes the identity map on the set $\{1,\ldots,n\}$.
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 $p$. Calculate the probability that $X_1, \ldots, X_n$ are all equal.
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 $p$. What is the distribution of $S = X_1 + \ldots + X_n$? A proof of the stated result is expected.
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 $i$ and $j$ be in $\{1, \ldots, n\}$. Give the distribution of the random variable $X_{i,j} = X_i \times X_j$.
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 $U(\omega) = (X_1(\omega), \ldots, X_n(\omega))^T$ and $M(\omega) = U(\omega)\,{}^t(U(\omega))$. Give the distribution, expectation and variance of the random variables $\operatorname{tr}(M)$ and $\operatorname{rg}(M)$.
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.
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?
Let $p \in ]0,1[$. We start from the zero matrix of $\mathcal{M}_n(\mathbb{R})$, denoted $M_0$. For all $k \in \mathbb{N}$, we construct the matrix $M_{k+1}$ from the matrix $M_k$ by scanning through the matrix in one pass and changing each zero coefficient to 1 with probability $p$, independently. For $k \geqslant 1$, the number of modifications made during the $k$-th pass is denoted $N_k$. Give the distribution of $N_1$, then the conditional distribution of $N_2$ given $(N_1 = i)$ for $i$ in a set to be specified. Are $N_1$ and $N_2$ independent?
Let $p \in ]0,1[$. We start from the zero matrix of $\mathcal{M}_n(\mathbb{R})$, denoted $M_0$. For all $k \in \mathbb{N}$, we construct the matrix $M_{k+1}$ from the matrix $M_k$ by scanning through the matrix in one pass and changing each zero coefficient to 1 with probability $p$, independently. We denote $q = 1-p$ and $m = n^2$. Let $i$ and $j$ be in $\{1, \ldots, n\}$. The smallest integer $k \geqslant 1$ such that the coefficient at row $i$, column $j$ of $M_k$ equals 1 is denoted $T_{i,j}$. Give the distribution of $T_{i,j}$.
Let $p \in ]0,1[$, $q = 1-p$, $m = n^2$. The smallest integer $k \geqslant 1$ such that the coefficient at row $i$, column $j$ of $M_k$ equals 1 is denoted $T_{i,j}$. For an integer $k \geqslant 1$, give the value of $P(T_{i,j} \geqslant k)$.
Let $p \in ]0,1[$, $q = 1-p$, $m = n^2$. For $k \geqslant 1$, the number of modifications made during the $k$-th pass is denoted $N_k$. Let $r \geqslant 1$ be an integer and $S_r = N_1 + \cdots + N_r$. What does $S_r$ represent? Give its distribution (you may use the previous question).
Let $p \in ]0,1[$, $q = 1-p$, $m = n^2$. We denote by $N$ the smallest index $k$ for which the matrix $M_k$ is completely filled. a) Propose an approach to approximate the expectation of $N$ using a computer simulation with the functions above. b) Give an expression for the exact value of this expectation involving $q$ and $m$.