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 $G$ be the set of elements $u \in E$ satisfying: (a) $u \in \operatorname{Im}(T)$, (b) $\forall v \in E, S(u,v) = 0$, where $S(v,w) = (v \mid T(w)) + (T(v) \mid w)$.
Deduce that the map $(v,w) \in G \times G \mapsto (T(v) \mid T(w))$ is a scalar product on $G$.

Let $A \in M _ { n } ( \mathbb { C } )$. Show that if $\rho ( A ) < 1$, then the sequence $\left( A ^ { k } \right) _ { k \in \mathbb { N } ^ { * } }$ converges to 0.

Let $m \geq 2$ be a natural integer and $E$ an $\mathbb{R}$-vector space of dimension $2m+1$. Let $T, M$ be two endomorphisms of $E$ satisfying (H1): $T^{2m} \neq 0_{\mathcal{L}(E)}$ and $T^{2m+1} = 0_{\mathcal{L}(E)}$, and (H4): $T \circ M + M \circ T = 0_{\mathcal{L}(E)}$.
Let $k \in \mathbb{N}$.
(a) Show that $M \circ T^k = (-1)^k T^k \circ M$.
(b) Deduce that $\operatorname{Im}(T^k)$ and $\operatorname{ker}(T^k)$ are stable under $M$.

Let $A \in M _ { n } ( \mathbb { C } )$. a) Show that, for all $k \in \mathbb { N } ^ { * } , \left\| A ^ { k } \right\| \geqslant \rho ( A ) ^ { k }$. b) We define the subset of $\mathbb { R } _ { + }$ $$E _ { A } = \left\{ \alpha > 0 \left\lvert \, \lim _ { k \rightarrow + \infty } \left( \frac { A } { \alpha } \right) ^ { k } = 0 \right. \right\} .$$ Show that $\left. E _ { A } = \right] \rho ( A ) , + \infty [$.

Let $m \geq 2$ be a natural integer and $E$ an $\mathbb{R}$-vector space of dimension $2m+1$. Let $T, M$ be two endomorphisms of $E$ satisfying (H1)–(H4). We set $F^+ = \operatorname{ker}(M - \operatorname{Id}_E)$, $F^- = \operatorname{ker}(M + \operatorname{Id}_E)$.
Show that one of the two following assertions is true: (i) $\operatorname{ker}(T) \subset F^+$, (ii) $\operatorname{ker}(T) \subset F^-$.

Let $A \in M _ { n } ( \mathbb { C } )$. Show the formula $$\lim _ { k \rightarrow + \infty } \left\| A ^ { k } \right\| ^ { 1 / k } = \rho ( A )$$

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 $T, M$ be two endomorphisms of $E$ satisfying (H1)–(H4). We set $F^+ = \operatorname{ker}(M - \operatorname{Id}_E)$, $F^- = \operatorname{ker}(M + \operatorname{Id}_E)$. Let $G$ be the set of elements $u \in E$ satisfying (a) $u \in \operatorname{Im}(T)$ and (b) $\forall v \in E, S(u,v) = 0$, where $S(v,w) = (v \mid T(w)) + (T(v) \mid w)$.
We assume here that $\operatorname{ker}(T) \subset F^+$.
(a) Show that $\forall z \in F^-, T^{2m}(z) = 0_E$.
(b) Show that $\operatorname{Im}(T)^\perp \subset F^+$ and that $\operatorname{Im}(T^2)^\perp \cap \operatorname{Im}(T) \subset F^-$.
(c) Let $z \in \operatorname{Im}(T)^\perp$ with $z \neq 0_E$. Show that $T(z) \in G^\perp$ and that $T(z) \neq 0_E$.
(d) Let $z \in \operatorname{Im}(T^2)^\perp \cap \operatorname{Im}(T)$ with $z \neq 0_E$. Show that $T(z) \in G^\perp$ and that $T(z) \neq 0_E$.

For $A \in M _ { n } ( \mathbb { C } )$ with coefficients $a _ { i , j }$, we set $A _ { + } = \left( b _ { i , j } \right) _ { 1 \leqslant i , j \leqslant n }$, where $b _ { i , j } = \left| a _ { i , j } \right|$. Show the inequality $$\rho ( A ) \leqslant \rho \left( A _ { + } \right)$$

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 $T, M$ be two endomorphisms of $E$ satisfying (H1)–(H4). We set $F^+ = \operatorname{ker}(M - \operatorname{Id}_E)$, $F^- = \operatorname{ker}(M + \operatorname{Id}_E)$. Let $G$ be the set of elements $u \in E$ satisfying (a) $u \in \operatorname{Im}(T)$ and (b) $\forall v \in E, S(u,v) = 0$.
We now say that a pair $(w_1, w_2) \in E \times E$ is a characterizing pair of $G$ if $w_1$ and $w_2$ satisfy the three properties:
(A) $w_1 \in F^+$, $T(w_1) \in G^\perp$ and $T(w_1) \neq 0_E$,
(B) $w_2 \in F^-$, $T(w_2) \in G^\perp$ and $T(w_2) \neq 0_E$,
(C) $w_i \in \operatorname{Im}(T^2)^\perp$ for $i = 1$ and $i = 2$.
Deduce from the previous questions the existence of a characterizing pair of $G$.

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 $T, M$ be two endomorphisms of $E$ satisfying (H1)–(H4). Let $G$ be the set of elements $u \in E$ satisfying (a) $u \in \operatorname{Im}(T)$ and (b) $\forall v \in E, S(u,v) = 0$.
Deduce that $\operatorname{dim}(G) \leq 2m-2$.

Let $x , y \in \mathbb { C } ^ { n } , \lambda , \mu \in \mathbb { C }$. Show that if $\lambda \neq \mu$, then the following implication holds $$\left( A x = \lambda x \quad \text { and } \quad { } ^ { t } A y = \mu y \right) \Longrightarrow { } ^ { t } x y = 0 .$$

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 $T, M$ be two endomorphisms of $E$ satisfying (H1)–(H4). We set $F^+ = \operatorname{ker}(M - \operatorname{Id}_E)$, $F^- = \operatorname{ker}(M + \operatorname{Id}_E)$. Let $G$ be the set of elements $u \in E$ satisfying (a) $u \in \operatorname{Im}(T)$ and (b) $\forall v \in E, S(u,v) = 0$.
We assume that $G$ satisfies hypothesis (H5): $\operatorname{dim}(G) = 2m-2$.
Show that if $(w_1, w_2)$ is a characterizing pair of $G$ then $(T(w_1), T(w_2))$ constitutes a basis of $G^\perp$.

Throughout this part, $A$ is a strictly positive matrix in $M _ { n } ( \mathbb { R } )$. Suppose that there exist a non-negative real $\mu$ and a positive non-zero vector $w$ such that $A w \geqslant \mu w$. a) Show that for all natural integer $k , A ^ { k } w \geqslant \mu ^ { k } w$. Deduce that $\rho ( A ) \geqslant \mu$. b) Show that if $A w > \mu w$, then $\rho ( A ) > \mu$. c) We now suppose that in the system of inequalities $A w \geqslant \mu w$, the $k$-th inequality is strict, that is $$\sum _ { j = 1 } ^ { n } a _ { k j } w _ { j } > \mu w _ { k } .$$ Show that there exists $\epsilon > 0$ such that, by setting $w _ { j } ^ { \prime } = w _ { j }$ if $j \neq k$ and $w _ { k } ^ { \prime } = w _ { k } + \epsilon$, we have $A w ^ { \prime } > \mu w ^ { \prime }$. Deduce that $\rho ( A ) > \mu$.

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 assume that $E$ is of dimension 4 and that $u$ has no real eigenvalue. Let $\mathcal{B}$ be a basis of $E$ such that $\operatorname{Mat}_{\mathcal{B}}(\omega) = J_4$, and let $U \in \mathcal{M}_4(\mathbb{R})$ be the matrix of $u$ in $\mathcal{B}$.
Show that $U$ is diagonalizable over $\mathbb { C }$. Deduce that there exist $\lambda \in \mathbb { C } \backslash \mathbb { R }$ and vectors $Z$ and $Y$ of $\mathbb { C } ^ { 4 }$ linearly independent over $\mathbb { C }$ such that $U Z = \lambda Z$ and $U Y = \overline{\lambda} Y$.

We keep all the notations from Part I and we assume that hypotheses (H1), (H2), (H3), (H4) and (H5) are all satisfied. Let $(w_1, w_2)$ be a characterizing pair of $G$ (satisfying properties (A), (B) and (C) of question 12). For any $\lambda \in \mathbb{R}$, we consider the following problem, denoted $\mathcal{P}_\lambda$: $$\text{Find } u \in G \text{ such that: } \forall v \in G, (u \mid v) - \lambda (T(u) \mid T(v)) = 0$$ and we denote by $H_\lambda$ the set of solutions $u$ of this problem.
(a) Show that if $(\mathcal{P}_\lambda)$ admits a solution $u \neq 0_E$, then necessarily $\lambda > 0$.
(b) Let $u \in G$. Show that $u$ is a solution of $(\mathcal{P}_\lambda)$ if and only if $$\left(\operatorname{Id}_E + \lambda T^2\right)(u) \in G^\perp$$ Deduce that there exist two real numbers $\alpha$ and $\beta$ such that: $$u = \alpha \left(\operatorname{Id}_E + \lambda T^2\right)^{-1} T(w_1) + \beta \left(\operatorname{Id}_E + \lambda T^2\right)^{-1} T(w_2)$$ (c) Show that the problem $(\mathcal{P}_\lambda)$ admits a non-zero solution if and only if $$Q_1(\lambda) \cdot Q_2(\lambda) = 0$$ where for $i \in \{1,2\}$, $Q_i$ is the polynomial $$Q_i(X) = \sum_{k=0}^{m-1} (-1)^k \left(T^{2k+1}(w_i) \mid T(w_i)\right) X^k$$ (d) Suppose that $\lambda$ is a root of the product polynomial $Q_1 Q_2$. Show that $\operatorname{dim}(H_\lambda) = 2$ if $\lambda$ is a common root of $Q_1$ and $Q_2$, and $\operatorname{dim}(H_\lambda) = 1$ otherwise.
(e) Show that $$\forall i \in \{1,2\}, Q_i(X) = \sum_{k=0}^{m-1} (-1)^k S\left(w_i, T^{2k+1}(w_i)\right) X^k$$

Throughout this part, $A$ is a strictly positive matrix in $M _ { n } ( \mathbb { R } )$. Let $\lambda$ be an eigenvalue of $A$ with modulus $\rho ( A )$ and let $x \in \mathbb { C } ^ { n } \backslash \{ 0 \}$ be an eigenvector of $A$ associated with $\lambda$. We define the positive non-zero vector $v _ { 0 }$ by $\left( v _ { 0 } \right) _ { i } = \left| x _ { i } \right|$ for $1 \leqslant i \leqslant n$. a) Show that $A v _ { 0 } \geqslant \rho ( A ) v _ { 0 }$, then that $$A v _ { 0 } = \rho ( A ) v _ { 0 }$$ b) Deduce that $\rho ( A ) > 0$ and $$\forall i \in \llbracket 1 , n \rrbracket , \left( v _ { 0 } \right) _ { i } > 0 .$$ c) Show that $x$ is collinear with $v _ { 0 }$. Deduce that $\lambda = \rho ( A )$.

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 assume that $E$ is of dimension 4 and that $u$ has no real eigenvalue. Let $\mathcal{B}$ be a basis of $E$ such that $\operatorname{Mat}_{\mathcal{B}}(\omega) = J_4$, and let $U \in \mathcal{M}_4(\mathbb{R})$ be the matrix of $u$ in $\mathcal{B}$. Let $\lambda \in \mathbb{C} \backslash \mathbb{R}$ and $Z, Y \in \mathbb{C}^4$ be as in question 15.
Let $Z _ { 1 } , Z _ { 2 } , Y _ { 1 } , Y _ { 2 }$ be vectors of $\mathbb { R } ^ { 4 }$ such that $Z = Z _ { 1 } + i Z _ { 2 }$ and $Y = Y _ { 1 } + i Y _ { 2 }$. Let $\left( z _ { 1 } , z _ { 2 } , y _ { 1 } , y _ { 2 } \right) \in E ^ { 4 }$ have coordinates respectively $Z _ { 1 } , Z _ { 2 } , Y _ { 1 } , Y _ { 2 }$ in the basis $\mathcal { B }$. Show that $\widetilde { \mathcal { B } } : = \left( z _ { 1 } , z _ { 2 } , y _ { 1 } , - y _ { 2 } \right)$ is a basis of $E$.

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 assume that $E$ is of dimension 4 and that $u$ has no real eigenvalue. Let $\mathcal{B}$ be a basis of $E$ such that $\operatorname{Mat}_{\mathcal{B}}(\omega) = J_4$, and let $z_1, z_2, y_1, y_2 \in E$ be as defined in question 16.
Show that $$\begin{aligned} & \omega \left( z _ { 1 } , z _ { 2 } \right) = \omega \left( y _ { 1 } , y _ { 2 } \right) = 0 \\ & \omega \left( z _ { 1 } , y _ { 1 } \right) = - \omega \left( z _ { 2 } , y _ { 2 } \right) \\ & \omega \left( z _ { 1 } , y _ { 2 } \right) = \omega \left( z _ { 2 } , y _ { 1 } \right) \end{aligned}$$

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 assume that $E$ is of dimension 4 and that $u$ has no real eigenvalue. Let $\mathcal{B}$ be a basis of $E$ such that $\operatorname{Mat}_{\mathcal{B}}(\omega) = J_4$, and let $Z, Y \in \mathbb{C}^4$ be eigenvectors as in question 15, with $z_1, z_2, y_1, y_2$ as defined in question 16.
Show that, by replacing $Y$ with $\xi Y$ where $\xi \in \mathbb { C } \backslash \{ 0 \}$ is suitably chosen, we have $\omega \left( z _ { 1 } , y _ { 1 } \right) = - 1$ and $\omega \left( z _ { 1 } , y _ { 2 } \right) = 0$.

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 assume that $E$ is of dimension 4 and that $u$ has no real eigenvalue. Let $\widetilde{\mathcal{B}} = (z_1, z_2, y_1, -y_2)$ be the basis constructed in questions 16--18, where $\operatorname{Mat}_{\widetilde{\mathcal{B}}}(\omega) = J_4$.
Show that there exist $r > 0$ and $\theta \in \mathbb { R } \backslash \pi \mathbb { Z }$ such that $$\operatorname { Mat } _ { \widetilde { \mathcal { B } } } ( u ) = r \left( \begin{array} { c c } R _ { \theta } & 0 \\ 0 & R _ { - \theta } \end{array} \right)$$ where $R _ { \theta } = \left( \begin{array} { c c } \cos \theta & - \sin \theta \\ \sin \theta & \cos \theta \end{array} \right)$, and conclude that $$\operatorname { Mat } _ { \widetilde { \mathcal { B } } } ( \omega ) = J _ { 4 } \quad \text{and} \quad \operatorname { Mat } _ { \widetilde { \mathcal { B } } } \left( \omega _ { 1 } \right) = r \left( \begin{array} { c c } 0 & - R _ { - \theta } \\ R _ { \theta } & 0 \end{array} \right).$$

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 $j = 1 , \ldots , r$.

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 $j = 1 , \ldots , r$. The notation $F^{\omega}$ is defined in question 7.
Show that, for all $j$ and $k$ belonging to $\{ 1 , \ldots , r \}$ and distinct, we have $F _ { k } \subset F _ { j } ^ { \omega }$ and $F _ { k } \subset F _ { j } ^ { \omega _ { 1 } }$.

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 $j = 1 , \ldots , r$, and $F_1, \ldots, F_r$ are pairwise orthogonal for $\omega$ and for $\omega_1$ (as shown in question 21).
Deduce that, for all $j \in \{ 1 , \ldots , r \}$, the restrictions of $\omega$ and $\omega _ { 1 }$ to $F _ { j } \times F _ { j }$ are symplectic forms on $F _ { j }$.

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 $(x,y) \in E^2$.
Suppose that the characteristic polynomial of $u$ has roots of multiplicity at most 2 in $\mathbb { C }$. Show that $E$ is the direct sum of subspaces of dimension 2 or 4, pairwise orthogonal for $\omega$ and $\omega _ { 1 }$, and on which the restrictions of $\omega$ and $\omega _ { 1 }$ are symplectic forms.

We fix two symplectic forms $\omega$ and $\omega _ { 1 }$ on $E$. We consider the propositions:
$\left( \mathcal { F } _ { 1 } \right) :$ There exists a complex structure tamed by $\omega$ and by $\omega _ { 1 }$.
$\left( \mathcal { F } _ { 2 } \right) :$ The segment $\left[ \omega , \omega _ { 1 } \right] = \left\{ ( 1 - \theta ) \omega + \theta \omega _ { 1 } ; \theta \in [ 0,1 ] \right\}$ is included in the set of symplectic forms on $E$.
Let $u$ be the automorphism of $E$ defined in question 13. Suppose that $\left( \mathcal { F } _ { 2 } \right)$ is satisfied and that the characteristic polynomial of $u$ has roots of multiplicity at most 2 in $\mathbb { C }$. Show that $( \mathcal { F } _ { 1 } )$ is satisfied.
Hint: You may prove and then use the fact that, for all $\theta \in \mathbb { R } \backslash \pi \mathbb { Z }$, there exists $\phi \in \mathbb { R }$ such that, for all $X \in \mathbb { R } ^ { 2 } \backslash \{ 0 \} , { } ^ { t } X R _ { \phi } X > 0$ and ${ } ^ { t } X R _ { \theta + \phi } X > 0$.