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.

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 $Z_n^{n^2-2n+2} = 2^{n-1} Z_n$ and recover the fact that $Z_n$ is not primitive.

Throughout this part, $A$ is a strictly positive matrix in $M _ { n } ( \mathbb { R } )$. We use the notation from question 17: $w_0$, $v_0$, $F = \left\{ x \in \mathbb { C } ^ { n } \mid { } ^ { t } x w _ { 0 } = 0 \right\}$, and $\mathbb { C } ^ { n } = F \oplus \mathbb { C } v _ { 0 }$. a) We denote by $\psi$ the endomorphism of $F$ defined as the restriction of $\varphi _ { A }$ to $F$. Show that all eigenvalues of $\psi$ have modulus strictly less than $\rho ( A )$. Deduce that $\rho ( A )$ is a simple root of the characteristic polynomial of $A$ and that $$\operatorname { ker } \left( A - \rho ( A ) I _ { n } \right) = \mathbb { C } v _ { 0 }$$ b) Show that if $x \in F , \lim _ { k \rightarrow + \infty } \frac { A ^ { k } x } { \rho ( A ) ^ { k } } = 0$. c) Let $x$ be a positive non-zero vector. Determine the limit of $\frac { A ^ { k } x } { \rho ( A ) ^ { k } }$ when $k$ tends to $+ \infty$.

Let $A \in \mathcal{M}_{n}(\mathbb{R})$ be a positively stable matrix. For all real $t$, we set $V(t) = \exp\left(-tA^{\top}\right) \exp(-tA)$ and $W(t) = \int_{0}^{t} V(s) \mathrm{d}s$.
a) Show that, for all real $t, V(t) \in \mathcal{S}_{n}^{++}(\mathbb{R})$ and that, if $t > 0, W(t) \in \mathcal{S}_{n}^{++}(\mathbb{R})$.
b) Show that, for all real $t, A^{\top} W(t) + W(t) A = I_{n} - V(t)$.
c) What do we obtain by letting $t$ tend to $+\infty$ in the previous equality? Deduce that the matrix $B$ of question III.C.2 is positive definite.

We assume that $p$ is an integer greater than or equal to 2, that $\left( a _ { k } \right) _ { k \in \mathbb { N } }$ and $\left( b _ { k } \right) _ { k \in \mathbb { N } }$ are two sequences of real numbers that are $p$-periodic and that $\forall k \in \mathbb { N } , b _ { k } \neq 0$. We denote by Sol(II.2) the set of complex sequences $\left( z _ { k } \right) _ { k \in \mathbb { N } }$ that satisfy the recurrence relation $$\forall k \in \mathbb { N } ^ { * } , \quad b _ { k } z _ { k + 1 } + a _ { k } z _ { k } + b _ { k - 1 } z _ { k - 1 } = 0$$ To any complex sequence $\left( z _ { k } \right) _ { k \in \mathbb { N } }$, we associate the sequence $\left( Z _ { k } \right) _ { k \in \mathbb { N } }$ of elements of $\mathbb { C } ^ { 2 }$ defined by $$\forall k \in \mathbb { N } , \quad Z _ { k } = \binom { z _ { k } } { z _ { k + 1 } }$$ Prove that the sequence $\left( z _ { k } \right) _ { k \in \mathbb { N } }$ is a solution of (II.2) if and only if the sequence $\left( Z _ { k } \right) _ { k \in \mathbb { N } }$ is a solution of a system (II.3) of the form $$\forall k \in \mathbb { N } , \quad Z _ { k + 1 } = A _ { k } Z _ { k }$$ Specify the matrix $A _ { k } \in \mathcal { M } _ { 2 } ( \mathbb { C } )$.

We denote $Q = A _ { p - 1 } A _ { p - 2 } \cdots A _ { 0 }$. We fix a solution $\left( Z _ { k } \right) _ { k \in \mathbb { N } }$ of (II.3). Prove that, for any natural integer $k$ and any natural integer $r \in \llbracket 1 , p - 1 \rrbracket$, $$\left\{ \begin{array} { l } Z _ { k p } = Q ^ { k } Z _ { 0 } \\ Z _ { k p + r } = A _ { r - 1 } A _ { r - 2 } \cdots A _ { 0 } Q ^ { k } Z _ { 0 } \end{array} \right.$$

We fix a natural number $p$ greater than or equal to 2. We recall the following result: for all matrices $A _ { 1 }$ and $A _ { 2 }$ in $\mathcal { M } _ { n } ( \mathbb { C } )$, all matrices $X _ { 1 }$ and $X _ { 2 }$ in $\mathcal { M } _ { n , 1 } ( \mathbb { C } )$ and all complex numbers $\lambda _ { 1 }$ and $\lambda _ { 2 }$: $$\left( \begin{array} { c c } A _ { 1 } & X _ { 1 } \\ 0 _ { 1 , n } & \lambda _ { 1 } \end{array} \right) \left( \begin{array} { c c } A _ { 2 } & X _ { 2 } \\ 0 _ { 1 , n } & \lambda _ { 2 } \end{array} \right) = \left( \begin{array} { c c } A _ { 1 } A _ { 2 } & A _ { 1 } X _ { 2 } + \lambda _ { 2 } X _ { 1 } \\ 0 _ { 1 , n } & \lambda _ { 1 } \lambda _ { 2 } \end{array} \right)$$ Let $A \in \mathcal { M } _ { n } ( \mathbb { C } ) , X \in \mathcal { M } _ { n , 1 } ( \mathbb { C } )$ and $\lambda \in \mathbb { C }$. Prove that, for all integer $k \geqslant 1$ we have: $$\left( \begin{array} { c c } A & X \\ 0 _ { 1 , n } & \lambda \end{array} \right) ^ { k } = \left( \begin{array} { c c } A ^ { k } & X _ { k } \\ 0 _ { 1 , n } & \lambda ^ { k } \end{array} \right)$$ where $X _ { k } = \left( \sum _ { j = 0 } ^ { k - 1 } \lambda ^ { k - 1 - j } A ^ { j } \right) X$.

We set $M_n = \left(\begin{array}{ccccc} 0 & 1 & 0 & \cdots & 0 \\ 0 & 0 & \ddots & \ddots & \vdots \\ \vdots & & \ddots & \ddots & 0 \\ 0 & & & \ddots & 1 \\ 1 & 0 & \cdots & \cdots & 0 \end{array}\right)$ and $\omega_n = \mathrm{e}^{2i\pi/n}$.
Calculate $M_n^2, \ldots, M_n^n$. Show that $M_n$ is invertible and give an annihilating polynomial of $M_n$.

We propose to study the matrix equation $R^2 = J_3$.
Let $R$ be a solution of this equation. Give the values of $R^4$ and $R^6$, then the set of solutions of the equation.

We fix a polynomial $f \in \mathbb{C}[X]$ of degree $n \geq 1$. We consider a complex number $z \in \overline{\mathbb{D}}$ and we define the matrices $M \in \mathcal{M}_{n+1}(\mathbb{C})$ and $P \in \mathcal{M}_{n+1,1}(\mathbb{C})$ by $$M = \left(\begin{array}{cccccc} z & 0 & 0 & \ldots & 0 & \sqrt{1-|z|^2} \\ \sqrt{1-|z|^2} & 0 & 0 & \ldots & 0 & -\bar{z} \\ 0 & & & & & 0 \\ 0 & & & & & 0 \\ \vdots & & I_{n-1} & & & \vdots \\ 0 & & & & & 0 \end{array}\right)$$ and $$P = \left(\begin{array}{c} 1 \\ 0 \\ \vdots \\ 0 \end{array}\right)$$ Show that $z^k = P^T M^k P$ for all integer $0 \leq k \leq n$.

In this part, we assume $n \geqslant 2$. For all $(a_{0}, \ldots, a_{n-1}) \in \mathbb{R}^{n}$, we set $$J(a_{0}, \ldots, a_{n-1}) = \left( \begin{array}{cccc} a_{0} & a_{n-1} & \cdots & a_{1} \\ a_{1} & a_{0} & \cdots & a_{2} \\ \vdots & \vdots & & \vdots \\ a_{n-1} & a_{n-2} & \cdots & a_{0} \end{array} \right)$$ The coefficient with index $(i,j)$ of $J(a_{0}, \ldots, a_{n-1})$ is $a_{i-j}$ if $i \geqslant j$ and $a_{i-j+n}$ if $i < j$. Let $\mathcal{A}$ be the set of matrices of $\mathcal{M}_{n}(\mathbb{R})$ of the form $J(a_{0}, \ldots, a_{n-1})$ where $(a_{0}, \ldots, a_{n-1}) \in \mathbb{R}^{n}$. Let $J \in \mathcal{M}_{n}(\mathbb{R})$ be the matrix canonically associated with the endomorphism $\varphi \in \mathcal{L}(\mathbb{R}^{n})$ defined by $\varphi: e_{j} \mapsto e_{j+1}$ if $j \in \{1, \ldots, n-1\}$ and $\varphi(e_{n}) = e_{1}$, where $(e_{1}, \ldots, e_{n})$ is the canonical basis of $\mathbb{R}^{n}$.
Specify the matrices $J$ and $J^{2}$. (One may distinguish the cases $n = 2$ and $n > 2$.)

In this part, we assume $n \geqslant 2$. For all $(a_{0}, \ldots, a_{n-1}) \in \mathbb{R}^{n}$, we set $$J(a_{0}, \ldots, a_{n-1}) = \left( \begin{array}{cccc} a_{0} & a_{n-1} & \cdots & a_{1} \\ a_{1} & a_{0} & \cdots & a_{2} \\ \vdots & \vdots & & \vdots \\ a_{n-1} & a_{n-2} & \cdots & a_{0} \end{array} \right)$$ The coefficient with index $(i,j)$ of $J(a_{0}, \ldots, a_{n-1})$ is $a_{i-j}$ if $i \geqslant j$ and $a_{i-j+n}$ if $i < j$. Let $\mathcal{A}$ be the set of matrices of $\mathcal{M}_{n}(\mathbb{R})$ of the form $J(a_{0}, \ldots, a_{n-1})$ where $(a_{0}, \ldots, a_{n-1}) \in \mathbb{R}^{n}$. Let $J \in \mathcal{M}_{n}(\mathbb{R})$ be the matrix canonically associated with the endomorphism $\varphi \in \mathcal{L}(\mathbb{R}^{n})$ defined by $\varphi: e_{j} \mapsto e_{j+1}$ if $j \in \{1, \ldots, n-1\}$ and $\varphi(e_{n}) = e_{1}$, where $(e_{1}, \ldots, e_{n})$ is the canonical basis of $\mathbb{R}^{n}$.
Specify the matrices $J^{n}$ and $J^{k}$ for $2 \leqslant k \leqslant n-1$.

We keep the notations from Parts II and III. We denote $r _ { k } = b - A x _ { k }$, $e _ { k } = x _ { k } - \tilde { x }$, and note that $r _ { k } = - A e _ { k }$. We denote by $m$ the smallest integer $k$ such that $H _ { k + 1 } = H _ { k }$.
Show that $e _ { k } \neq 0$ for $k \in \{ 0 , \ldots , m - 1 \}$, and that $e _ { k } = 0$ for $k \geq m$.

Let $E$ be a $\mathbb{C}$-vector space of dimension $n \geq 1$. Let $u$ be a diagonalizable endomorphism of $E$. Show that $u$ is a permutation endomorphism if and only if there exist natural integers $c_1, \ldots, c_n$ such that, for all $k \in \mathbb{N}$,
$$\operatorname{Tr}\left(u^k\right) = \sum_{\substack{\ell=1 \\ \ell \mid k}}^{n} \ell c_\ell$$
(We sum over the values of $\ell$ dividing $k$ and belonging to $\llbracket 1, n \rrbracket$.)

Throughout part II, $A \in \mathcal{M}_n(\mathbb{R})$ is a strictly positive matrix. Show that $\forall k \in \mathbb{N}^*, A^k > 0$.

Throughout part II, $A \in \mathcal{M}_n(\mathbb{R})$ is a strictly positive matrix. Assume $A$ is diagonalizable over $\mathbb{C}$. Show that, if $\rho(A) < 1$ then $\lim_{k \rightarrow +\infty} A^k = 0$.

Throughout part II, $A \in \mathcal{M}_n(\mathbb{R})$ is a strictly positive matrix satisfying $\rho(A) = 1$. We consider an eigenvalue $\lambda \in \mathbb{C}$ of $A$ with modulus 1 and $x$ an eigenvector associated with $\lambda$. We assume that $|x| < A|x|$ and that there exists $\varepsilon > 0$ such that $A^2|x| - A|x| > \varepsilon A|x|$. We set $B = \frac{1}{1+\varepsilon} A$. Show that for all $k \geqslant 1, B^k A|x| \geqslant A|x|$.

Throughout part II, $A \in \mathcal{M}_n(\mathbb{R})$ is a strictly positive matrix satisfying $\rho(A) = 1$. We set $B = \frac{1}{1+\varepsilon} A$ for some $\varepsilon > 0$. Determine $\lim_{k \rightarrow +\infty} B^k$.

We assume that $A$ is strictly positive and diagonalizable over $\mathbb{C}$. For all $Y \in \mathcal{M}_{n,1}(\mathbb{R})$, for all $p \in \mathbb{N}^*$, we denote $Y_p = \left(\frac{A}{\rho(A)}\right)^p Y$. Let $\lambda \in S = \operatorname{sp}(A) \setminus \{\rho(A)\}$. Let $Y \in \ker\left(A - \lambda I_n\right)$. Show that the sequence $\left(Y_p\right)_{p \in \mathbb{N}^*}$ converges to 0.

We assume that $A$ is strictly positive and diagonalizable over $\mathbb{C}$. For all $Y \in \mathcal{M}_{n,1}(\mathbb{R})$, for all $p \in \mathbb{N}^*$, we denote $Y_p = \left(\frac{A}{\rho(A)}\right)^p Y$. Let $Y \in \mathcal{M}_{n,1}(\mathbb{R})$ be a positive vector. Show that the sequence $\left(Y_p\right)_{p \in \mathbb{N}^*}$ converges to the projection of $Y$ onto $E_{\rho(A)}(A)$ parallel to $\bigoplus_{\lambda \in S} E_\lambda(A)$. Verify that, if it is non-zero, this latter vector (the projection of $Y$) is strictly positive.

Show that $\lim_{k \rightarrow +\infty} \frac{\operatorname{tr}\left(A^{k+1}\right)}{\operatorname{tr}\left(A^k\right)} = \rho(A)$.

Throughout part III, $N$ is a fixed non-zero natural integer and $(X_n)_{n \in \mathbb{N}}$ is a homogeneous Markov chain on $\llbracket 0, N \rrbracket$ with transition matrix $Q$ where $q_{i,j} = P(X_{n+1} = j \mid X_n = i) > 0$. We define $a_{i,j}(t) = q_{i,j} \mathrm{e}^{jt}$, $A(t) = \left(a_{i,j}(t)\right)_{0 \leqslant i,j \leqslant N}$, $z_j(t) = P(X_1 = j)\mathrm{e}^{jt}$, $Z(t) = \begin{pmatrix} z_0(t) \\ \vdots \\ z_N(t) \end{pmatrix}$, and $Y^{(n)}(t) = (A(t))^{n-1} Z(t)$ so that $E\left(\mathrm{e}^{tS_n}\right) = \sum_{j=0}^N Y_j^{(n)}(t)$. Let $\gamma(t)$ be the dominant eigenvalue of $A(t)$. Show that $\lim_{n \rightarrow +\infty} \frac{\ln\left(E\left(\mathrm{e}^{tS_n}\right)\right)}{n} = \lambda(t)$ where $\lambda(t) = \ln(\gamma(t))$.