We denote $\| u \| = \sup _ { \substack { x \in E \\ x \neq 0 } } \frac { \| u ( x ) \| } { \| x \| }$. Show that $\|.\|$ is a norm on $\mathcal { L } ( E )$.

Show that $S_n^+(\mathrm{R})$ and $S_n^{++}(\mathrm{R})$ are convex subsets of $M_n(\mathrm{R})$. Are they vector subspaces of $M_n(\mathrm{R})$?

Let $\mathscr{P}$ be the set of row vectors of size $d$ with non-negative coefficients whose coordinate sum equals 1: $$\mathscr{P} = \left\{ u \in \mathscr{M}_{1,d}\left(\mathbb{R}_{+}\right) : \sum_{j=1}^{d} u_j = 1 \right\}.$$ We consider a square matrix $P \in \mathscr{M}_d\left(\mathbb{R}_{+}\right)$ such that for all $i \in \{1,\ldots,d\}$, $$\sum_{j=1}^{d} P_{i,j} = 1$$ We further assume that there exist $\nu \in \mathscr{P}$ and $c > 0$ such that for all $i,j \in \{1,\ldots,d\}$, $$P_{i,j} \geqslant c\nu_j.$$
Show that if $u \in \mathscr{P}$, then $uP \in \mathscr{P}$.

Show that it is a sub-multiplicative norm, that is: $$\forall ( u , v ) \in \mathcal { L } ( E ) ^ { 2 } , \quad \| u v \| \leqslant \| u \| \cdot \| v \| ,$$ and deduce a bound on $\left\| u ^ { k } \right\|$, for any natural number $k$, in terms of $\| u \|$ and the integer $k$.

Show that, if $A \in S_n^{++}(\mathrm{R})$, there exists $S \in S_n^{++}(\mathrm{R})$ such that $A = S^2$.

Let $\mathscr{P}$ be the set of row vectors of size $d$ with non-negative coefficients whose coordinate sum equals 1: $$\mathscr{P} = \left\{ u \in \mathscr{M}_{1,d}\left(\mathbb{R}_{+}\right) : \sum_{j=1}^{d} u_j = 1 \right\}.$$ We consider a square matrix $P \in \mathscr{M}_d\left(\mathbb{R}_{+}\right)$ such that for all $i \in \{1,\ldots,d\}$, $$\sum_{j=1}^{d} P_{i,j} = 1$$ We further assume that there exist $\nu \in \mathscr{P}$ and $c > 0$ such that for all $i,j \in \{1,\ldots,d\}$, $$P_{i,j} \geqslant c\nu_j.$$
Show that for all $u, v \in \mathscr{P}$, $$\|uP - vP\|_1 \leqslant (1-c)\|u - v\|_1.$$ (One may introduce $R = P - cN$ where $N = (n_{i,j})_{1 \leqslant i,j \leqslant d}$ with $n_{i,j} = \nu_j$ for all $1 \leqslant i,j \leqslant d$.)

Let $n$ be a non-zero natural number.

1. Let $p$ be a vector projection of rank $r \in \mathbb { N }$.
   1. [1.1.] Give, as a function of $r$, a matrix $W$ of $p$ in an adapted basis.
   2. [1.2.] Give the possible spectra of $W$.
   3. [1.3.] Compare $\boldsymbol { \operatorname { rg } } ( W )$ and $\boldsymbol { \operatorname { tr } } ( W )$.
   4. [1.4.] Calculate $\boldsymbol { \operatorname { det } } ( W )$.

We consider the family $X _ { 1 } , \ldots , X _ { n }$ of independent random variables defined on the same probability space $( \Omega , \mathscr { A } , \mathbb { P } )$ all following the Bernoulli distribution with parameter $p \in ]0,1[$.
Let $M$ be a discrete random variable from $\Omega$ to $\mathscr { M } _ { n } ( \mathbb { R } )$ such that for all $\omega$ in $\Omega , M ( \omega )$ is diagonalisable and similar to $\Delta ( \omega ) = \operatorname { diag } \left( X _ { 1 } ( \omega ) , \ldots , X _ { n } ( \omega ) \right)$.

1. We denote by $T$ the random variable $\mathbf { tr } ( M )$.
   1. [2.1.] Determine $T ( \Omega )$, that is the set of values taken by the random variable $T$.
   2. [2.2.] Give the probability distribution of $T$ and the expectation of the random variable $T$.
2. Deduce the probability distribution of the random variable $R = \mathbf { rg } ( M )$.
3. We denote by $D$ the random variable $\boldsymbol { \operatorname { det } } ( M )$.
   1. [4.1.] Determine $D ( \Omega )$.
   2. [4.2.] Give the probability distribution of $D$ and calculate the expectation of the random variable $D$.
4. We propose to determine the probability of the event $Z$: ``the eigenspaces of the matrix $M$ all have the same dimension''.
   1. [5.1.] We denote by $V$ the event: ``$M$ has only one eigenvalue''. Calculate $\mathbb { P } ( V )$.
   2. [5.2.] Suppose $n$ is odd. Determine $\mathbb { P } ( Z )$.
   3. [5.3.] Suppose $n$ is even and set $n = 2 r$. Calculate $\mathbb { P } ( T = r )$. Deduce $\mathbb { P } ( Z )$.
5. For all $\omega \in \Omega$, we denote $U ( \omega ) = \left( \begin{array} { c } X _ { 1 } ( \omega ) \\ \vdots \\ X _ { n } ( \omega ) \end{array} \right) \in \mathscr { M } _ { n , 1 } ( \mathbb { R } )$ and $A ( \omega ) = U ( \omega ) \times ( U ( \omega ) ) ^ { \top } = \left( a _ { i j } ( \omega ) \right) _ { ( i , j ) \in \llbracket 1 , n \rrbracket ^ { 2 } }$.
   1. [6.1.] Let $\omega \in \Omega$. Determine, for all pairs $( i , j ) \in \llbracket 1 , n \rrbracket ^ { 2 } , a _ { i j } ( \omega )$.
   2. [6.2.] Give the probability distribution of each random variable $a _ { i j }$.
   3. [6.3.] Show that $\operatorname { tr } ( A ) = \sum _ { i = 1 } ^ { n } X _ { i }$.
   4. [6.4.] Determine the values taken by the random variable $\boldsymbol { \operatorname { rg } } ( A )$.
   5. [6.5.] For all $\omega$ in $\Omega$, give the eigenvalues of the matrix $A ( \omega )$.
   6. [6.6.] Determine the probability distribution of the random variable $\mathbf { rg } ( A )$.

Let $\mathscr{P}$ be the set of row vectors of size $d$ with non-negative coefficients whose coordinate sum equals 1: $$\mathscr{P} = \left\{ u \in \mathscr{M}_{1,d}\left(\mathbb{R}_{+}\right) : \sum_{j=1}^{d} u_j = 1 \right\}.$$ We consider a square matrix $P \in \mathscr{M}_d\left(\mathbb{R}_{+}\right)$ such that for all $i \in \{1,\ldots,d\}$, $$\sum_{j=1}^{d} P_{i,j} = 1$$ We further assume that there exist $\nu \in \mathscr{P}$ and $c > 0$ such that for all $i,j \in \{1,\ldots,d\}$, $$P_{i,j} \geqslant c\nu_j.$$
Let $(x_n)_n \in \mathscr{P}^{\mathbb{N}}$ be defined by recursion by $x_0 \in \mathscr{P}$ and $$x_{n+1} = x_n P.$$ Show that the series $\sum_{n \geqslant 0} \|x_{n+1} - x_n\|_1$ is convergent.

Let $\mathscr{P}$ be the set of row vectors of size $d$ with non-negative coefficients whose coordinate sum equals 1: $$\mathscr{P} = \left\{ u \in \mathscr{M}_{1,d}\left(\mathbb{R}_{+}\right) : \sum_{j=1}^{d} u_j = 1 \right\}.$$ We consider a square matrix $P \in \mathscr{M}_d\left(\mathbb{R}_{+}\right)$ such that for all $i \in \{1,\ldots,d\}$, $$\sum_{j=1}^{d} P_{i,j} = 1$$ We further assume that there exist $\nu \in \mathscr{P}$ and $c > 0$ such that for all $i,j \in \{1,\ldots,d\}$, $$P_{i,j} \geqslant c\nu_j.$$ Let $(x_n)_n \in \mathscr{P}^{\mathbb{N}}$ be defined by recursion by $x_0 \in \mathscr{P}$ and $x_{n+1} = x_n P$.
Deduce that $(x_n)_n$ converges to an element of $\mathscr{P}$.

Let $\mathscr{P}$ be the set of row vectors of size $d$ with non-negative coefficients whose coordinate sum equals 1: $$\mathscr{P} = \left\{ u \in \mathscr{M}_{1,d}\left(\mathbb{R}_{+}\right) : \sum_{j=1}^{d} u_j = 1 \right\}.$$ We consider a square matrix $P \in \mathscr{M}_d\left(\mathbb{R}_{+}\right)$ such that for all $i \in \{1,\ldots,d\}$, $$\sum_{j=1}^{d} P_{i,j} = 1$$ We further assume that there exist $\nu \in \mathscr{P}$ and $c > 0$ such that for all $i,j \in \{1,\ldots,d\}$, $$P_{i,j} \geqslant c\nu_j.$$
Show that there exists a unique element $\mu$ of $\mathscr{P}$ such that $\mu P = \mu$.

Let $\mathscr{P}$ be the set of row vectors of size $d$ with non-negative coefficients whose coordinate sum equals 1: $$\mathscr{P} = \left\{ u \in \mathscr{M}_{1,d}\left(\mathbb{R}_{+}\right) : \sum_{j=1}^{d} u_j = 1 \right\}.$$ We consider a square matrix $P \in \mathscr{M}_d\left(\mathbb{R}_{+}\right)$ such that for all $i \in \{1,\ldots,d\}$, $$\sum_{j=1}^{d} P_{i,j} = 1$$ We further assume that there exist $\nu \in \mathscr{P}$ and $c > 0$ such that for all $i,j \in \{1,\ldots,d\}$, $$P_{i,j} \geqslant c\nu_j.$$ Let $\mu$ be the unique element of $\mathscr{P}$ such that $\mu P = \mu$.
Show that for all $n \in \mathbb{N}$ and all $x \in \mathscr{P}$, $$\left\| xP^n - \mu \right\|_1 \leqslant 2(1-c)^n.$$

Let $M \in \mathscr{M}_d\left(\mathbb{R}_{+}\right)$. We assume that the matrix $M$ has an eigenvalue $\lambda > 0$ and that there exists $h \in \mathscr{M}_{d,1}\left(\mathbb{R}_{+}^{*}\right)$ a column vector such that: $$Mh = \lambda h.$$ We also assume that there exist $\nu \in \mathscr{P}$ and $c > 0$ such that for all $i,j \in \{1,\ldots,d\}$, $$M_{i,j} \geqslant c\nu_j.$$ We introduce the matrix $P \in \mathscr{M}_d\left(\mathbb{R}_{+}\right)$ defined for $1 \leqslant i,j \leqslant d$ by $$P_{i,j} = \frac{M_{i,j} h_j}{\lambda h_i}.$$
Justify that for all $i \in \{1,\ldots,d\}$, $\displaystyle\sum_{j=1}^{d} P_{i,j} = 1$.

Let $M \in \mathscr{M}_d\left(\mathbb{R}_{+}\right)$. We assume that the matrix $M$ has an eigenvalue $\lambda > 0$ and that there exists $h \in \mathscr{M}_{d,1}\left(\mathbb{R}_{+}^{*}\right)$ a column vector such that: $$Mh = \lambda h.$$ We also assume that there exist $\nu \in \mathscr{P}$ and $c > 0$ such that for all $i,j \in \{1,\ldots,d\}$, $$M_{i,j} \geqslant c\nu_j.$$ We introduce the matrix $P \in \mathscr{M}_d\left(\mathbb{R}_{+}\right)$ defined for $1 \leqslant i,j \leqslant d$ by $$P_{i,j} = \frac{M_{i,j} h_j}{\lambda h_i}.$$
Let $n \geqslant 1$. Give an expression for the coefficients of $P^n$ in terms of the coefficients of $M^n$, $h$ and $\lambda$.

Let $M \in \mathscr{M}_d\left(\mathbb{R}_{+}\right)$. We assume that the matrix $M$ has an eigenvalue $\lambda > 0$ and that there exists $h \in \mathscr{M}_{d,1}\left(\mathbb{R}_{+}^{*}\right)$ a column vector such that: $$Mh = \lambda h.$$ We also assume that there exist $\nu \in \mathscr{P}$ and $c > 0$ such that for all $i,j \in \{1,\ldots,d\}$, $$M_{i,j} \geqslant c\nu_j.$$ We introduce the matrix $P \in \mathscr{M}_d\left(\mathbb{R}_{+}\right)$ defined for $1 \leqslant i,j \leqslant d$ by $$P_{i,j} = \frac{M_{i,j} h_j}{\lambda h_i}.$$
(a) Show that there exist $\mu \in \mathscr{P}$, $C > 0$ and $\gamma \in [0,1[$, such that $\mu P = \mu$ and for all $n \geqslant 0$, $$\sum_{i=1}^{d} \sum_{j=1}^{d} \left| \lambda^{-n} \left(M^n\right)_{i,j} - h_i \frac{\mu_j}{h_j} \right| \leqslant C\gamma^n.$$
(b) Prove that there exists a unique $\pi \in \mathscr{P}$ such that $\pi M = \lambda \pi$.

We consider a Markov kernel $K$. We assume that 1 is a simple eigenvalue of $K$. We assume that there exists a probability $\pi \in \mathscr{M}_{1,N}(\mathbf{R})$ such that:
(a) For all $j \in \llbracket 1;N \rrbracket$, $\pi[j] \neq 0$.
(b) $\forall (i,j) \in \llbracket 1;N \rrbracket^2$, $\pi[i] K[i,j] = K[j,i] \pi[j]$; we say that $K$ is $\pi$-reversible. Show that $\pi K = \pi$.

For $X, Y \in \mathscr{M}_{N,1}(\mathbf{R})^2$, we define $$\langle X, Y \rangle = \sum_{i=1}^{N} X[i] Y[i] \pi[i]$$ where $\pi \in \mathscr{M}_{1,N}(\mathbf{R})$ is a probability with $\pi[j] \neq 0$ for all $j$. Show that $(X, Y) \mapsto \langle X, Y \rangle$ is an inner product on $\mathscr{M}_{N,1}(\mathbf{R})$.

Consider $a = (a_1, \ldots, a_d) \in \left(\mathbb{R}_{+}^{*}\right)^d$ and $b = (b_1, \ldots, b_{d-1}) \in \left(\mathbb{R}_{+}^{*}\right)^{d-1}$ and introduce the matrix $$M = \begin{pmatrix} a_1 & b_1 & 0 & \ldots & 0 & 0 \\ a_2 & 0 & b_2 & \ldots & 0 & 0 \\ a_3 & 0 & 0 & \ldots & 0 & 0 \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\ a_{d-1} & 0 & 0 & \ldots & 0 & b_{d-1} \\ a_d & 0 & 0 & \ldots & 0 & 0 \end{pmatrix}.$$
(a) Justify that there exists a unique pair $(\lambda, \pi) \in \mathbb{R}_{+}^{*} \times \mathscr{P}$ such that $\pi M = \lambda \pi$. Express $\pi$ explicitly in terms of $a$, $b$ and $\lambda$.
(b) Show that there exists a unique $h \in \mathscr{M}_{d,1}\left(\mathbb{R}_{+}^{*}\right)$ such that $\langle \pi, h \rangle = 1$ and $$Mh = \lambda h.$$
(c) Deduce that the sequence $\left(\lambda^{-n} M^n\right)_{n \geqslant 1}$ converges as $n$ tends to infinity and give an expression for its limit in terms of $h$ and $\mu$.

We consider the Euclidean space $E = \mathscr{M}_{N,1}(\mathbf{R})$ equipped with the inner product $\langle X, Y \rangle = \sum_{i=1}^{N} X[i] Y[i] \pi[i]$, where $\pi$ is a $\pi$-reversible probability for the Markov kernel $K$. We consider the endomorphism of $E$ defined by $u : X \mapsto (I_N - K)X$. Show that $\ker(u) = \operatorname{Vect}(U)$ and that $u$ is a self-adjoint endomorphism of $E$.

We assume that for all $i,j \in \{1,\ldots,d\}$, $N_{i,j}$ is a random variable taking values in $\mathbb{N}$ and such that $N_{i,j}^2$ has finite expectation. For all $i \in \{1,\ldots,d\}$, we introduce the random variable $L_i = (N_{i,1}, \ldots, N_{i,d})$. We consider a family of independent random variables $(L_i^{n,k})_{n \geqslant 1, k \geqslant 1}$ where for all $i$, $n$, $k$, $L_i^{n,k}$ has the same distribution as $L_i$. Let $X_0 = (X_{0,i})_{1 \leqslant i \leqslant d}$ be a random variable taking values in $\mathscr{M}_{1,d}(\mathbb{N})$. We define by recursion for $n \geqslant 0$: $$X_{n+1} = \sum_{i=1}^{d} \sum_{k=1}^{X_{n,i}} L_i^{n,k}.$$ We introduce $M \in \mathscr{M}_d\left(\mathbb{R}_{+}\right)$ defined by $M_{i,j} = \mathbb{E}(N_{i,j})$ and $x_{n,j} = \mathbb{E}(X_{n,j})$.
(a) Show that, for all $y \in \mathscr{M}_{1,d}(\mathbb{N})$ and $1 \leqslant j \leqslant d$, $$\mathbb{E}\left(X_{n+1,j} \mathbf{1}_{X_n = y}\right) = (yM)_j \mathbb{P}(X_n = y).$$ (One may use without proof the fact that the random variables $L_i^{n,k}$ and $\mathbf{1}_{X_n = y}$ are independent.)
(b) Deduce that, for all $n \geqslant 0$, $$x_{n+1} = x_n M.$$

We consider the Euclidean space $E = \mathscr{M}_{N,1}(\mathbf{R})$ equipped with the inner product $\langle X, Y \rangle = \sum_{i=1}^{N} X[i] Y[i] \pi[i]$, and the endomorphism $u : X \mapsto (I_N - K)X$ with $q_u(X) = (u(X) \mid X)$. Show that for all $X \in E$, $$q_u(X) = \frac{1}{2} \sum_{i=1}^{N} \sum_{j=1}^{N} (X[i] - X[j])^2 K[i,j] \pi[i]$$ What can be said about the eigenvalues of $u$?

We denote $\|\cdot\|_2$ the canonical Euclidean norm on $\mathbb{R}^2$ and we denote $$\mathcal{C} := \left\{ x \in \mathbb{R}^2 \mid \|x\|_2 = 1 \right\}$$ We fix an arbitrary norm $\|\cdot\|$ on $\mathbb{R}^2$ and we denote $$\mathcal{K} = \left\{ A \in M_2(\mathbb{R}) \mid \forall x \in \mathbb{R}^2,\ \|x\|_2 \geq \|Ax\| \right\}.$$ a) Show that $\mathcal{K}$ is a compact and convex subset of $M_2(\mathbb{R})$. b) Show that there exists $A \in \mathcal{K}$ such that $\det A = \sup_{B \in \mathcal{K}} \det B$.

Let $\mathscr{I}$ be a finite set and $(Y_i)_{i \in \mathscr{I}}$ be a family of random variables that are pairwise independent, take real values and whose squares have finite expectation. Show that $$\mathbb{E}\left(\left(\sum_{i \in \mathscr{I}} Y_i\right)^2\right) = \left(\sum_{i \in \mathscr{I}} \mathbb{E}(Y_i)\right)^2 + \sum_{i \in \mathscr{I}} \operatorname{Var}(Y_i).$$

We consider the Euclidean space $E = \mathscr{M}_{N,1}(\mathbf{R})$ equipped with the inner product $\langle X, Y \rangle = \sum_{i=1}^{N} X[i] Y[i] \pi[i]$, and the matrix $H_t$ defined by $\forall (i,j) \in \llbracket 1;N \rrbracket^2, H_t[i,j] = e^{-t} \sum_{n=0}^{+\infty} \frac{t^n K^n[i,j]}{n!}$. Let $X \in E$. We denote by $\psi_X$ the function defined from $\mathbf{R}$ to $E$ by $\psi_X : t \mapsto H_t X$ and $\varphi_X$ the function defined from $\mathbf{R}$ to $\mathbf{R}$ by $\varphi_X : t \mapsto \|H_t X\|^2$. Justify that $\psi_X$ is differentiable and that for all $t$ in $\mathbf{R}$, $$\psi_X'(t) = -(I_N - K) H_t X$$

We fix an element $A$ of $\mathcal{K}$ such that $\det A = \sup_{B \in \mathcal{K}} \det B$. Show that $\det A > 0$ and that there exists $x \in \mathcal{C}$ such that $\|Ax\| = 1$.

We use the setup of the third part. For $u \in \mathscr{M}_{d,1}(\mathbb{R})$, we denote $T(u) = (T_i(u))_{1 \leqslant i \leqslant d} \in \mathbb{R}^d$ the vector defined by $$T_i(u) = \operatorname{Var}\left(\langle L_i, u \rangle\right) \quad \text{for } i \in \{1,\ldots,d\}.$$
(a) Show that for all $u \in \mathscr{M}_{d,1}(\mathbb{R})$, $y \in \mathscr{M}_{1,d}(\mathbb{N})$ and $n \geqslant 0$, $$\mathbb{E}\left(\langle X_{n+1}, u \rangle^2 \mathbf{1}_{X_n = y}\right) = \mathbb{P}(X_n = y)\left(\langle y, Mu \rangle^2 + \langle y, T(u) \rangle\right).$$ (One may use without proof the fact that, for all $n \geqslant 0$, the random variables $\sum_{j=1}^{d} u_j L_{i,j}^{n,k} \mathbf{1}_{X_n = y}$ are pairwise independent when $k$ and $i$ vary.)
(b) Show that for all $u \in \mathscr{M}_{d,1}(\mathbb{R})$ and $n \geqslant 0$, $$\mathbb{E}\left(\langle X_{n+1}, u \rangle^2\right) = \mathbb{E}\left(\langle X_n, Mu \rangle^2\right) + \langle x_0 M^n, T(u) \rangle.$$