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 consider $M \in \mathscr { D } _ { \rho } \left( S _ { n } ( \mathbb { R } ) \right)$. We fix a real eigenvalue $\lambda$ of $M _ { \mid t = 0 }$ and denote by $d$ its multiplicity as a root of $\chi _ { \mid t = 0 }$ where $\chi = \det(XI_n - M)$. We set $A = F ( M )$ and $B = G ( M )$; we thus have $A , B \in \mathscr { D } _ { \rho _ { 1 } } \left( S _ { n } ( \mathbb { R } ) \right)$. For $a \in U _ { \rho _ { 1 } }$, we set $A _ { a } = A _ { \mid t = a }$ and $B _ { a } = B _ { \mid t = a }$.
Show that there exist two matrices $U \in \mathscr { M } _ { n , d } ( \mathbb { R } )$ and $V \in \mathscr { M } _ { n , n - d } ( \mathbb { R } )$ such that:

• $\operatorname { im } \left( B _ { 0 } U \right) = \operatorname { im } \left( B _ { 0 } \right)$,
• $\operatorname { im } \left( A _ { 0 } V \right) = \operatorname { im } \left( A _ { 0 } \right)$ and
• the block matrix $\left( B _ { 0 } U \mid A _ { 0 } V \right)$ is invertible.

We consider $M \in \mathscr { D } _ { \rho } \left( S _ { n } ( \mathbb { R } ) \right)$. We fix a real eigenvalue $\lambda$ of $M _ { \mid t = 0 }$ and denote by $d$ its multiplicity as a root of $\chi _ { \mid t = 0 }$ where $\chi = \det(XI_n - M)$. We set $A = F ( M )$, $B = G ( M )$, $Q = ( B U \mid A V ) \in \mathscr { D } _ { \rho _ { 2 } } \left( \mathscr { M } _ { n } ( \mathbb { R } ) \right)$. For $a \in U _ { \rho _ { 2 } }$, we set $A _ { a } = A _ { \mid t = a }$ and $B _ { a } = B _ { \mid t = a }$. We consider a real number $a \in U _ { \rho _ { 2 } }$.
22a. Show that $\operatorname { im } \left( B _ { a } U \right) \oplus \operatorname { im } \left( A _ { a } V \right) = \mathbb { R } ^ { n }$.
22b. Show the equalities:

• $\operatorname { im } \left( B _ { a } U \right) = \operatorname { im } \left( B _ { a } \right) = \operatorname { ker } \left( A _ { a } \right)$ and
• $\operatorname { im } \left( A _ { a } V \right) = \operatorname { im } \left( A _ { a } \right) = \operatorname { ker } \left( B _ { a } \right)$.

(One may begin by showing the inclusions from left to right, then use a dimension argument.)

We consider $M \in \mathscr { D } _ { \rho } \left( S _ { n } ( \mathbb { R } ) \right)$. We fix a real eigenvalue $\lambda$ of $M _ { \mid t = 0 }$ and denote by $d$ its multiplicity as a root of $\chi _ { \mid t = 0 }$ where $\chi = \det(XI_n - M)$. We set $A = F ( M )$, $B = G ( M )$, $Q = ( B U \mid A V ) \in \mathscr { D } _ { \rho _ { 2 } } \left( \mathscr { M } _ { n } ( \mathbb { R } ) \right)$.
Show that $Q ^ { - 1 } \cdot M \cdot Q = \operatorname { Diag } \left( M _ { 1 } , M _ { 2 } \right)$ with $M _ { 1 } \in \mathscr { D } _ { \rho _ { 2 } } \left( \mathscr { M } _ { d } ( \mathbb { R } ) \right) , M _ { 2 } \in \mathscr { D } _ { \rho _ { 2 } } \left( \mathscr { M } _ { n - d } ( \mathbb { R } ) \right)$.

Let $f$ be a symmetric endomorphism of $\mathbf{R}^n$ with eigenvalues $\lambda_1 \leqslant \ldots \leqslant \lambda_n$ and associated orthonormal eigenbasis $(e_1, \ldots, e_n)$.
Let $k \in \llbracket 1, n \rrbracket$ and $S_k$ a vector subspace of $\mathbf{R}^n$ of dimension $k$. We set $T_k = \operatorname{Vect}(e_k, \ldots, e_n)$.
Justify that $S_k \cap T_k \neq \{0\}$.

Show that $\mathbb{M}_n(u) \neq \{0_n\}$.

In this part, we assume that $n \geqslant 4$. Let $u = (u_k)_{k \geqslant 0}$ be a sequence of $\mathbb{C}$ satisfying the condition $(C^\star)$: $R_u > 1$.
Let $H \in \mathscr{M}_n(\mathbb{C})$ be the matrix given by $$H = \begin{pmatrix} 0 & 1 & 0 & \cdots & 0 \\ 0 & 0 & 1 & \ddots & \vdots \\ \vdots & \ddots & \ddots & \ddots & 0 \\ \vdots & & \ddots & \ddots & 1 \\ 0 & \cdots & \cdots & 0 & 0 \end{pmatrix}.$$
(a) Determine the polynomial $\varphi_H$ in this case.
(b) Let $A = H + \alpha I_n$ where $\alpha \in \mathbb{C}$ is such that $|\alpha| < R_u$. Show that $$u(A) = \sum_{k=0}^{n-1} \frac{U^{(k)}(\alpha)}{k!} H^k$$ and deduce that $$u(A) = \begin{pmatrix} U(\alpha) & \frac{U^{(1)}(\alpha)}{1!} & \frac{U^{(2)}(\alpha)}{2!} & \cdots & \frac{U^{(n-1)}(\alpha)}{(n-1)!} \\ 0 & U(\alpha) & \frac{U^{(1)}(\alpha)}{1!} & \ddots & \vdots \\ \vdots & \ddots & \ddots & \ddots & \frac{U^{(2)}(\alpha)}{2!} \\ \vdots & & \ddots & \ddots & \frac{U^{(1)}(\alpha)}{1!} \\ 0 & \cdots & \cdots & 0 & U(\alpha) \end{pmatrix}.$$

Let $m \in \mathbb{N}$. We consider the matrix $$M = \left(\begin{array}{cccccc} \binom{0}{0} & 0 & \cdots & \cdots & \cdots & 0 \\ \binom{1}{0} & \binom{1}{1} & 0 & & & \vdots \\ \vdots & & \ddots & \ddots & & \vdots \\ \vdots & & & \ddots & \ddots & \vdots \\ \binom{m-1}{0} & & & & \binom{m-1}{m-1} & 0 \\ \binom{m}{0} & \cdots & \cdots & \cdots & \cdots & \binom{m}{m} \end{array}\right) \in \mathscr{M}_{m+1}(\mathbb{R}).$$ Justify that the families $\left(1, X, \ldots, X^{m}\right)$ and $\left(1, (X-1), \ldots, (X-1)^{m}\right)$ are bases of $\mathbb{R}_{m}[X]$.

Let $m \in \mathbb{N}$. We consider the matrix $$M = \left(\begin{array}{cccccc} \binom{0}{0} & 0 & \cdots & \cdots & \cdots & 0 \\ \binom{1}{0} & \binom{1}{1} & 0 & & & \vdots \\ \vdots & & \ddots & \ddots & & \vdots \\ \vdots & & & \ddots & \ddots & \vdots \\ \binom{m-1}{0} & & & & \binom{m-1}{m-1} & 0 \\ \binom{m}{0} & \cdots & \cdots & \cdots & \cdots & \binom{m}{m} \end{array}\right) \in \mathscr{M}_{m+1}(\mathbb{R}).$$ Show that the transpose of $M$ is the matrix of the linear map identity $$\begin{array}{ccc} \mathbb{R}_{m}[X] & \longrightarrow & \mathbb{R}_{m}[X] \\ P & \longmapsto & P \end{array}$$ in the bases $\left(1, X, \ldots, X^{m}\right)$ at the start and $\left(1, (X-1), \ldots, (X-1)^{m}\right)$ at the end.

Let $m \in \mathbb{N}$. We consider the matrix $$M = \left(\begin{array}{cccccc} \binom{0}{0} & 0 & \cdots & \cdots & \cdots & 0 \\ \binom{1}{0} & \binom{1}{1} & 0 & & & \vdots \\ \vdots & & \ddots & \ddots & & \vdots \\ \vdots & & & \ddots & \ddots & \vdots \\ \binom{m-1}{0} & & & & \binom{m-1}{m-1} & 0 \\ \binom{m}{0} & \cdots & \cdots & \cdots & \cdots & \binom{m}{m} \end{array}\right) \in \mathscr{M}_{m+1}(\mathbb{R}).$$ Justify that the families $\left(1, X, \ldots, X^m\right)$ and $\left(1, (X-1), \ldots, (X-1)^m\right)$ are bases of $\mathbb{R}_m[X]$.

Let $m \in \mathbb{N}$. We consider the matrix $$M = \left(\begin{array}{cccccc} \binom{0}{0} & 0 & \cdots & \cdots & \cdots & 0 \\ \binom{1}{0} & \binom{1}{1} & 0 & & & \vdots \\ \vdots & & \ddots & \ddots & & \vdots \\ \vdots & & & \ddots & \ddots & \vdots \\ \binom{m-1}{0} & & & & \binom{m-1}{m-1} & 0 \\ \binom{m}{0} & \cdots & \cdots & \cdots & \cdots & \binom{m}{m} \end{array}\right) \in \mathscr{M}_{m+1}(\mathbb{R}).$$ Show that the transpose of $M$ is the matrix of the linear map identity $$\begin{array}{ccc} \mathbb{R}_m[X] & \longrightarrow & \mathbb{R}_m[X] \\ P & \longmapsto & P \end{array}$$ in the bases $\left(1, X, \ldots, X^m\right)$ at the start and $\left(1, (X-1), \ldots, (X-1)^m\right)$ at the end.

Let $E$ be a $\mathbb{C}$-vector space of dimension $n$. Let $g$ be an endomorphism of $E$ such that $g^n = 0$ and $g^{n-1} \neq 0$.
Prove that there exists a basis of $E$ in which the matrix of $g$ is the matrix $N$ below: $$N = \left(\begin{array}{ccccc} 0 & 1 & 0 & \ldots & 0 \\ 0 & 0 & 1 & \ddots & \vdots \\ \vdots & \ddots & \ddots & \ddots & 0 \\ \vdots & & & \ddots & 1 \\ 0 & \ldots & & \ldots & 0 \end{array}\right)$$ In other words: $N = (n_{i,j})_{1 \leq i,j \leq n}$ with $n_{i,j} = 1$ if $j = i+1$ and $n_{i,j} = 0$ otherwise.

For every polynomial $P = P(X) \in \mathbf{C}_{n-1}[X]$ we set $$\left\{\begin{array}{l} s_1(P) = P(-X) \\ s_2(P) = P(1-X) \\ g(P) = P(X+1) - P(X) \end{array}\right.$$ We thus define three endomorphisms of the vector space $\mathbb{C}_{n-1}[X]$.
Calculate $s_1^2$, $s_2^2$ and express $s_1 \circ s_2$ in terms of $g$ and $Id_{\mathbb{C}_{n-1}[X]}$.

Let $E$ be a $\mathbf{C}$-vector space of dimension $n$. Let $g$ be an endomorphism of $E$ such that $g^n = 0$ and $g^{n-1} \neq 0$. Prove that there exists a basis of $E$ in which the matrix of $g$ is the matrix $N$ below: $$N = \left(\begin{array}{ccccc} 0 & 1 & 0 & \ldots & 0 \\ 0 & 0 & 1 & \ddots & \vdots \\ \vdots & \ddots & \ddots & \ddots & 0 \\ \vdots & & & \ddots & 1 \\ 0 & \ldots & & \ldots & 0 \end{array}\right)$$ In other words: $N = (n_{i,j})_{1 \leq i,j \leq n}$ with $n_{i,j} = 1$ if $j = i+1$ and $n_{i,j} = 0$ otherwise.

Until the end of part B, we assume that no root of $p$ is stable.
For every $j \in \llbracket 1, n \rrbracket$, we define the rational function $g_j \in E$ by $$g_j = \frac{f_j}{\prod_{i=1}^{n}(1 - \alpha_i X)}$$ and the map $P_j$, which associates to a rational function $f \in E$ the rational function $$P_j(f) = \frac{(1 - \alpha_j X)f - (1 - \alpha_j^2)f(\alpha_j)}{X - \alpha_j}$$
Show that for every $j \in \llbracket 1, n \rrbracket$, the map $P_j$ is an endomorphism of $E$ and determine its kernel.

Until the end of part B, we assume that no root of $p$ is stable.
For every $j \in \llbracket 1, n \rrbracket$ and every $g \in E$, compute $P_j\left(\frac{(X - \alpha_j)g}{1 - \alpha_j X}\right)$.

Until the end of part B, we assume that no root of $p$ is stable.
Deduce that the family $(f_1, \ldots, f_n)$ is linearly independent.

Show that the family $\left((S^\top)^i U\right)_{0 \leq i \leq n-1}$ is a basis of $\mathcal{M}_{n,1}(\mathbf{R})$. The matrices $S$ and $U$ were defined in the preliminary part of the problem.

Show, using questions 9 and 13, that if $p$ has no stable root and if $J(p)$ is not invertible then there exists a non-zero polynomial $q$ with real coefficients of degree at most $n-1$ such that $q(S^\top) U = 0_{n,1}$.

A matrix invariant For a square matrix $M$ and a nonzero natural integer $k$, we denote $$\delta_k(M) = -\operatorname{dim}\ker M^{k-1} + 2\operatorname{dim}\ker M^k - \operatorname{dim}\ker M^{k+1}.$$
a) Prove that if two square matrices $M$ and $M'$ are similar, then $\delta_k(M) = \delta_k(M')$ for all $k$.
b) Let $r$ be a nonzero natural integer. Verify that for all nonzero integer $k$, $\delta_k(J_r)$ equals 1 if $k = r$ and 0 otherwise.
c) Let $M_1$ and $M_2$ be two square matrices and let $M = \operatorname{diag}(M_1, M_2)$. Prove the relation $\operatorname{dim}\ker M = \operatorname{dim}\ker M_1 + \operatorname{dim}\ker M_2$ and then that for all nonzero integer $k$, $$\delta_k(M) = \delta_k(M_1) + \delta_k(M_2).$$ You may use without proof the fact that all these relations extend to a block diagonal matrix $\operatorname{diag}(M_1, \ldots, M_s)$.

The linear application $\widehat{\xi}$ and the endomorphism $\xi$ We denote by $\widehat{\xi} : \mathbb{C}[X^{\pm 1}] \rightarrow \mathcal{D}$ the linear application that to a Laurent polynomial $F$ associates $$\widehat{\xi}(F) = \Pi(XF) \quad \text{and} \quad \xi = \widehat{\xi}_{\mathcal{D}}$$ that is the endomorphism of $\mathcal{D}$ induced by $\widehat{\xi}$.
a) Let $F$ be an element of $\mathbb{C}[X^{\pm 1}]$. Prove that $\widehat{\xi}(\Pi(F)) = \widehat{\xi}(F)$.
b) Let $P$ be a polynomial and let $F$ be an element of $\mathcal{D}$. Prove that $P(\xi)(F) = \Pi(PF)$.

Image and kernel of powers of $\xi$ Let $n$ be a natural integer. Prove that $\xi^n$ is surjective and give a basis of the kernel of $\xi^n$.

Cyclic subspaces Let $r$ be a nonzero natural integer. Prove that the smallest vector subspace $\mathcal{D}_r$ of $\mathcal{D}$ containing $X^{-r}$ and stable by $\xi$ admits as basis $(X^{k-r})_{0 \leqslant k \leqslant r-1}$. Write the matrix of the endomorphism $\xi_{\mathcal{D}_r}$ induced by $\xi$ on $\mathcal{D}_r$ in this basis.

Compatible extension with $u$ given by a vector Let $V$ be a finite-dimensional vector space equipped with a nilpotent endomorphism $u$. We assume that there exists a vector subspace $W$ of $V$ stable by $u$ and a linear application $\varphi : W \rightarrow \mathcal{D}$ such that $\xi \circ \varphi = \varphi \circ u_W$. In this question, we assume that $W$ is strictly contained in $V$ and we fix a vector $v$ of $V$ that does not belong to $W$.
a) Verify that the set $$\mathcal{J} = \{P \in \mathbb{C}[X],\, P(u)(v) \in W\}$$ is an ideal of $\mathbb{C}[X]$.
b) Prove that there exists a natural integer $n$ such that $X^n \in \mathcal{J}$. Deduce that $\mathcal{J}$ is generated by the monomial $X^r$ for an appropriate natural integer $r$ that we do not ask you to specify.
c) Let $W'$ be the subspace of $V$ defined by $$W' = \{P(u)(v) + w,\, P \in \mathbb{C}[X] \text{ and } w \in W\}.$$ Verify that $W'$ contains $W$ and $v$ and that it is stable by $u$.
We denote $G_v = \varphi(u^r(v))$.
d) Prove that there exists an element $F_v$ of $\mathcal{D}$ such that $$G_v = \xi^r(F_v).$$
e) Let $P$ be a polynomial and let $w$ be an element of $W$. Prove that if $P(u)(v) = w$, then $P(\xi)(F_v) = \varphi(w)$.
f) Let $x$ be an element of $W'$. Let $P$ be a polynomial and let $w$ be an element of $W$ such that $x = P(u)(v) + w$. Prove that the element $\varphi'(x) = P(\xi)(F_v) + \varphi(w)$ depends only on $x$ and not on the choice of $P$ and $w$. Verify then that the application $\varphi'$ thus defined is an extension of $\varphi$ to $W'$ compatible with $u$ (it is not asked to verify that $\varphi'$ is linear, which we will admit).

Extension to $V$ compatible with $u$ Let $V$ be a finite-dimensional vector space equipped with a nilpotent endomorphism $u$. We assume that there exists a vector subspace $W$ of $V$ stable by $u$ and a linear application $\varphi : W \rightarrow \mathcal{D}$ such that $\xi \circ \varphi = \varphi \circ u_W$. Prove that $\varphi$ admits an extension $\psi$ to $V$ compatible with $u$.