In this part, $n$ is a non-zero natural integer, $M$ is in $\mathcal{M}_n(\mathbb{R})$ and $f$ is the endomorphism of $E = \mathcal{M}_{n,1}(\mathbb{R})$ defined by $f(X) = MX$ for all $X$ in $E$.
If we denote $X_i = \begin{pmatrix} \delta_{1,i} \\ \vdots \\ \delta_{n,i} \end{pmatrix}$ where $\delta_{k,l} = \begin{cases} 1 & \text{if } k = l \\ 0 & \text{if } k \neq l \end{cases}$ and $\mathcal{B}_n = (X_i)_{1 \leqslant i \leqslant n}$ the canonical basis of $E$, what is the matrix of $f$ in $\mathcal{B}_n$?

In this part $E$ is a real vector space of dimension $n$ equipped with a basis $\mathcal{B} = (\varepsilon_i)_{1 \leqslant i \leqslant n}$. We consider an endomorphism $f$ of $E$ and we denote by $A$ its matrix in the basis $\mathcal{B}$. There exists a unique inner product on $E$ for which $\mathcal{B}$ is orthonormal, denoted $\langle u, v \rangle$ or $u \cdot v$.
Let $H$ be a hyperplane of $E$ and $D$ its orthogonal complement. If $(u)$ is a basis of $D$ and if $U$ is the column matrix of $u$ in $\mathcal{B}$, show that $H$ is stable by $f$ if and only if $U$ is an eigenvector of the transpose of $A$.

Is the map $s^{\downarrow}$ linear? Justify your answer.

For $m \in \mathbb{R}$, we denote by $\mathcal{M}$ the matrix $$\mathcal{M} = \left(\begin{array}{ccc} 0 & -m & 0 \\ m & 0 & 0 \\ 0 & 0 & 0 \end{array}\right)$$
Show that for $X \in \mathbb{R}^{3}$, $\mathcal{M}X \cdot X = 0$. Geometrically interpret the application $\mathbb{R}^{3} \rightarrow \mathbb{R}^{3},\, X \mapsto (I_{3} + \mathcal{M})X$.

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.

For all $\alpha$ in $\mathbb{R}$ and for all distinct integers $i$ and $j$ between 1 and $n$, describe the effect on a square matrix $M$ of size $n$ of left multiplication by $I_n + \alpha E_{ij}$. Same question for right multiplication.

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 $j \in \mathbb{Z}$?

We denote $E_{n} = \mathcal{M}_{n,1}(\mathbb{R})$ equipped with the inner product $(X \mid Y) = X^{\top}Y$. A matrix $K \in \mathcal{M}_{n}(\mathbb{R})$ is called $F$-singular if there exists a non-zero $X \in F$ such that $\forall Z \in F, Z^{\top}KX = 0$. We assume $n \geqslant 2$. Let $F = H$ be a hyperplane of $E_{n}$ and let $N \in E_{n}$ be a unit vector normal to $H$.
Show that $A$ is $H$-singular if and only if there exist a non-zero vector $X$ in $H$ and a real number $\lambda$ such that $AX = \lambda N$.

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 $$\forall k \in \mathbb { N } ^ { * } , \quad b _ { k } z _ { k + 1 } + a _ { k } z _ { k } + b _ { k - 1 } z _ { k - 1 } = 0$$ Justify that the application $\left. \Psi : \left\lvert \, \begin{array} { l l l } \operatorname { Sol } ( \mathrm { II } .2 ) & \rightarrow & \mathbb { C } ^ { 2 } \\ \left( z _ { k } \right) _ { k \in \mathbb { N } } & \mapsto & \left( z _ { 0 } \right) \\ z _ { 1 } \end{array} \right. \right)$ is an isomorphism of $\mathbb { C }$-vector spaces.

Let $n$ and $p$ be two integers greater than or equal to 2. We fix throughout this part a sequence $\left( A _ { k } \right) _ { k \in \mathbb { N } }$ of matrices of $\mathrm { GL } _ { n } ( \mathbb { C } )$ which we assume to be $p$-periodic, that is such that $\forall k \in \mathbb { N } , A _ { k + p } = A _ { k }$. We denote by $\operatorname { Sol }$ (III.1) the set of sequences $\left( Y _ { k } \right) _ { k \in \mathbb { N } }$ of vectors of $\mathbb { C } ^ { n }$ satisfying the recurrence relation $$\forall k \in \mathbb { N } , \quad Y _ { k + 1 } = A _ { k } Y _ { k }$$ Justify that we define a sequence $\left( \Phi _ { k } \right) _ { k \in \mathbb { N } }$ of matrices of $\mathrm { GL } _ { n } ( \mathbb { C } )$ by setting $\left\{ \begin{array} { l } \Phi _ { 0 } = I _ { n } \\ \Phi _ { k + 1 } = A _ { k } \Phi _ { k } \quad \forall k \in \mathbb { N } \end{array} \right.$ and that $\left( Y _ { k } \right) _ { k \in \mathbb { N } } \in \operatorname { Sol }$ (III.1) if and only if $\forall k \in \mathbb { N } , Y _ { k } = \Phi _ { k } Y _ { 0 }$.

Show that the dimension of the vector space $E ^ { * }$ equals $n$.

We fix two symplectic forms $\omega$ and $\omega _ { 1 }$ on $E$, and let $u \in \mathrm{GL}(E)$ be the unique automorphism such that $\omega_1(x,y) = \omega(u(x),y)$ for all $(x,y) \in E^2$. We consider a polynomial $P \in \mathbb { R } [ X ]$ annihilating $u$ and a decomposition $P = P _ { 1 } \cdots P _ { r }$, where $r \in \mathbb { N } ^ { * }$ and $P _ { 1 } , \ldots , P _ { r }$ are polynomials pairwise coprime in $\mathbb { R } [ X ]$. We denote $F _ { j } = \operatorname { ker } \left[ P _ { j } ( u ) \right]$ for $j = 1 , \ldots , r$.
Show that $E = F _ { 1 } \oplus \cdots \oplus F _ { r }$ and that $F _ { j }$ is stable under $u$ for $j = 1 , \ldots , r$.

Let $m \geq 2$ be a natural integer and $E$ an $\mathbb{R}$-vector space of dimension $2m+1$. This space is equipped with a scalar product $(.|.)$. Let $T, M$ be two endomorphisms of $E$ satisfying the following hypotheses: (H1) $T^{2m} \neq 0_{\mathcal{L}(E)}$ and $T^{2m+1} = 0_{\mathcal{L}(E)}$. (H2) $M^2 = \operatorname{Id}_E$. (H3) $\forall (v,w) \in E^2, (M(v) \mid w) = (v \mid M(w))$. (H4) $T \circ M + M \circ T = 0_{\mathcal{L}(E)}$. We set $F^+ = \operatorname{ker}(M - \operatorname{Id}_E)$, $F^- = \operatorname{ker}(M + \operatorname{Id}_E)$.
For any vector $v \in E$, we set $$v^+ = v + M(v), \quad v^- = v - M(v)$$ (a) Show that $\forall v \in E, v^+ \in F^+$ and $v^- \in F^-$.
(b) Show that $E = F^+ \oplus^\perp F^-$.
(c) Show that $\forall v \in F^+, T(v) \in F^-$ and that $\forall v \in F^-, T(v) \in F^+$.
Deduce that $F^+$ and $F^-$ are stable under $T^2$.

Let $m \geq 2$ be a natural integer and $E$ an $\mathbb{R}$-vector space of dimension $2m+1$. Let $T$ be an endomorphism of $E$ satisfying (H1): $T^{2m} \neq 0_{\mathcal{L}(E)}$ and $T^{2m+1} = 0_{\mathcal{L}(E)}$.
Show that for all $k \in \{0, 1, \ldots, 2m\}$, $\operatorname{Im}(T^{k+1}) \subset \operatorname{Im}(T^k)$ and $\operatorname{Im}(T^{k+1}) \neq \operatorname{Im}(T^k)$.

Let $m \geq 2$ be a natural integer and $E$ an $\mathbb{R}$-vector space of dimension $2m+1$. Let $T$ be an endomorphism of $E$ satisfying (H1): $T^{2m} \neq 0_{\mathcal{L}(E)}$ and $T^{2m+1} = 0_{\mathcal{L}(E)}$.
Deduce that for all $k \in \{0, \ldots, 2m+1\}$, we have $$\operatorname{dim}(\operatorname{Im}(T^k)) = 2m+1-k, \quad \operatorname{dim}(\operatorname{ker}(T^k)) = k$$

Let $m \geq 2$ be a natural integer and $E$ an $\mathbb{R}$-vector space of dimension $2m+1$. Let $T$ be an endomorphism of $E$ satisfying (H1): $T^{2m} \neq 0_{\mathcal{L}(E)}$ and $T^{2m+1} = 0_{\mathcal{L}(E)}$.
Deduce also that $\operatorname{Im}(T^k) = \operatorname{ker}(T^{2m+1-k})$ for $0 \leq k \leq 2m+1$.

Let $m \geq 2$ be a natural integer and $E$ an $\mathbb{R}$-vector space of dimension $2m+1$ equipped with a scalar product $(.|.)$. Let $T$ be an endomorphism of $E$ satisfying (H1): $T^{2m} \neq 0_{\mathcal{L}(E)}$ and $T^{2m+1} = 0_{\mathcal{L}(E)}$.
Let $k \in \{1, 2, \ldots, 2m+1\}$ and $z \in \operatorname{Im}(T^k)^\perp \cap \operatorname{Im}(T^{k-1})$ such that $z \neq 0_E$. After justifying the existence of such a vector $z$, show that $T^{2m+1-k}(z) \neq 0_E$.

Let $m \geq 2$ be a natural integer and $E$ an $\mathbb{R}$-vector space of dimension $2m+1$ equipped with a scalar product $(.|.)$. Let $T$ be an endomorphism of $E$ satisfying (H1): $T^{2m} \neq 0_{\mathcal{L}(E)}$ and $T^{2m+1} = 0_{\mathcal{L}(E)}$. We consider the map $S$ from $E \times E$ to $\mathbb{R}$ defined by $$\forall (v,w) \in E^2, S(v,w) = (v \mid T(w)) + (T(v) \mid w)$$ and we denote by $G$ the set of elements $u \in E$ satisfying: (a) $u \in \operatorname{Im}(T)$, (b) $\forall v \in E, S(u,v) = 0$.
Show that $G$ is a vector subspace of $E$ and that $G \cap \operatorname{ker}(T) = \{0_E\}$.

Show that if $M$ is in $\mathcal{M}_n(\mathbb{C})$, then the following propositions are equivalent:
i. there exists $x_0$ in $\mathbb{C}^n$ such that $(x_0, f_M(x_0), \ldots, f_M^{n-1}(x_0))$ is a basis of $\mathbb{C}^n$;
ii. $M$ is similar to the matrix $C(a_0, \ldots, a_{n-1})$ defined by $$C(a_0, \ldots, a_{n-1}) = \left(\begin{array}{ccccc} 0 & 0 & \cdots & 0 & a_0 \\ 1 & \ddots & & \vdots & a_1 \\ 0 & \ddots & \ddots & \vdots & \vdots \\ \vdots & \ddots & \ddots & 0 & \vdots \\ 0 & \cdots & 0 & 1 & a_{n-1} \end{array}\right)$$ where $(a_0, \ldots, a_{n-1})$ are complex numbers.

Let $M$ be a cyclic matrix and $x_0$ be a cyclic vector of $f_M$. Let $P \in \mathbb{C}[X]$. Show that $P(f_M) \in \mathcal{C}(f_M)$, where $\mathcal{C}(f_M) = \{g \in \mathcal{L}(\mathbb{C}^n) \mid f_M \circ g = g \circ f_M\}$.

Let $M$ be a cyclic matrix and $x_0$ be a cyclic vector of $f_M$. Let $g \in \mathcal{C}(f_M)$. Show that there exist $(\alpha_0, \ldots, \alpha_{n-1}) \in \mathbb{C}^n$ such that $g = \alpha_0 Id_{\mathbb{C}^n} + \alpha_1 f_M + \cdots + \alpha_{n-1} f_M^{n-1}$. One may use the basis $(x_0, f_M(x_0), \ldots, f_M^{n-1}(x_0))$ and express $g(x_0)$ in this basis.

Let $M$ be a cyclic matrix and $x_0$ be a cyclic vector of $f_M$. The set $\mathcal{C}(f_M) = \{g \in \mathcal{L}(\mathbb{C}^n) \mid f_M \circ g = g \circ f_M\}$ is sought to be shown to be the set of polynomials in $f_M$. Conclude.

Let $N = \left(\begin{array}{ccccc} 0 & 0 & \cdots & \cdots & 0 \\ 1 & 0 & & & \vdots \\ 0 & \ddots & \ddots & & \vdots \\ \vdots & \ddots & \ddots & \ddots & \vdots \\ 0 & \cdots & 0 & 1 & 0 \end{array}\right)$.
Is the matrix $N$ cyclic?

We define the operators $$\mathcal{S}: \begin{cases} \mathcal{M}_n(\mathbb{R}) \rightarrow \mathcal{M}_n(\mathbb{R}) \\ X \mapsto NX - XN \end{cases} \quad \text{and} \quad \mathcal{S}^*: \begin{cases} \mathcal{M}_n(\mathbb{R}) \rightarrow \mathcal{M}_n(\mathbb{R}) \\ X \mapsto {}^t N X - X {}^t N \end{cases}$$ Show that the kernel of $\mathcal{S}$ is the set of real Toeplitz matrices that are lower triangular. We admit that the kernel of $\mathcal{S}^*$ is the set of real Toeplitz matrices that are upper triangular.

Show that $\mathcal{S}(\Delta_{k+1}) \subset \Delta_k$ and $\mathcal{S}^*(\Delta_k) \subset \Delta_{k+1}$.