Say, by briefly justifying the answer, whether the following assertions are correct for all $A , B \in M _ { n } ( \mathbb { C } ) , \mu \in \mathbb { C }$. i) $\rho ( \mu A ) = | \mu | \rho ( A )$. ii) $\rho ( A + B ) \leqslant \rho ( A ) + \rho ( B )$. iii) $\rho ( A B ) \leqslant \rho ( A ) \rho ( B )$. iv) For $P \in M _ { n } ( \mathbb { C } )$ invertible, $\rho \left( P ^ { - 1 } A P \right) = \rho ( A )$. v) $\rho \left( { } ^ { t } A \right) = \rho ( A )$.

We fix a symplectic form $\omega$ on $E$. Let $F$ be a vector subspace of $E$.
(a) Show that, for every linear form $u : F \rightarrow \mathbb { R }$, there exists a linear form $\widetilde { u } : E \rightarrow \mathbb { R }$ whose restriction to $F$ coincides with $u$.
We denote by $F ^ { \omega }$ the vector subspace of $E$ defined by $$F ^ { \omega } = \{ x \in E : \forall y \in F , \omega ( x , y ) = 0 \}$$ and $\psi _ { F }$ the linear map defined by $$\left\lvert \, \begin{aligned} \psi _ { F } : \quad E & \rightarrow F ^ { * } \\ x & \left. \mapsto \varphi _ { \omega } ( x ) \right| _ { F } \end{aligned} \right.$$ where $\left. \varphi _ { \omega } ( x ) \right| _ { F }$ is the restriction of $\varphi _ { \omega } ( x )$ to $F$.
(b) Show that the restriction of $\omega$ to $F \times F$ is a symplectic form on $F$ if and only if $F \cap F ^ { \omega } = \{ 0 \}$.
(c) What are the kernel and image of $\psi _ { F }$ ?
(d) Show that $\operatorname { dim } ( F ) + \operatorname { dim } \left( F ^ { \omega } \right) = \operatorname { dim } ( E )$.
(e) Show that, if the restriction of $\omega$ to $F \times F$ is a symplectic form on $F$, then $E = F \oplus F ^ { \omega }$ and the restriction of $\omega$ to $F ^ { \omega } \times F ^ { \omega }$ is a symplectic form on $F ^ { \omega }$.

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 for any matrix $A \in M _ { n } ( \mathbb { C } )$, $$\rho ( A ) \leqslant \| A \| .$$

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 $T$ 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.

We fix a symplectic form $\omega$ on $E$. Conclude that there exists a basis $\mathcal { B }$ of $E$ such that $\operatorname { Mat_{\mathcal {B}} } ( \omega ) = J _ { n }$. Deduce that $\omega$ tames at least one complex structure on $E$.

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

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 $$\rho ( A ) \leqslant \rho \left( A _ { + } \right)$$

We fix two symplectic forms $\omega$ and $\omega _ { 1 }$ on $E$. Show that there exists a unique $u \in \mathrm { GL } ( E )$ such that $\omega _ { 1 } ( x , y ) = \omega ( u ( x ) , y )$ for all $( x , y ) \in E ^ { 2 }$. Show then that $u$ belongs to the set $\mathcal { S }$ defined by $$\mathcal { S } = \left\{ u \in \mathrm { GL } ( E ) : \forall ( x , y ) \in E ^ { 2 } , \omega ( x , u ( y ) ) = \omega ( u ( x ) , y ) \right\}$$

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

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. Let $\mathcal { B }$ be a basis of $E$ such that $\operatorname { Mat } _ { \mathcal { B } } ( \omega ) = J _ { 4 }$. Let $U \in \mathcal { M } _ { 4 } ( \mathbb { R } )$ be the matrix of $u$ in the basis $\mathcal { B }$.
(a) What relation is there between the matrices $J _ { 4 }$ and $U$ ?
(b) Show that there exist $N \in \mathcal { M } _ { 2 } ( \mathbb { R } )$ and $\alpha , \beta \in \mathbb { R }$ such that $$U = \left( \begin{array} { c c } N & \alpha J _ { 2 } \\ \beta J _ { 2 } & { } ^ { t } N \end{array} \right)$$
(c) Determine, as a function of $N , \alpha$ and $\beta$, the coefficients of the polynomial $T$ defined by $T ( X ) = \operatorname { det } \left( N - X I _ { 2 } \right) + \alpha \beta$. Show that $T$ is an annihilating polynomial of $U$.

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 $\rho ( A ) > \mu$.

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 keep all the notations from Parts I and II and assume hypotheses (H1)–(H5). Let $\mathcal{B} = (z_1, \ldots, z_\ell)$, where $\ell = 2m-2$, be a basis of $G$. For any element $u$ of $G$, we denote by $U$ (capital letter) the column vector containing the coordinates of $u$ with respect to the basis $\mathcal{B}$. We denote by $A = [a_{i,j}]_{1 \leq i,j \leq \ell}$ and $B = [b_{i,j}]_{1 \leq i,j \leq \ell}$ the two square matrices whose coefficients are defined by $$\forall 1 \leq i,j \leq \ell, \quad a_{i,j} = (z_i \mid z_j), \quad b_{i,j} = (T(z_i) \mid T(z_j))$$
(a) Let $u, v \in G$. Show that $$(u \mid v) = {}^t U A V, \quad (T(u) \mid T(v)) = {}^t U B V$$ and deduce that $A$ and $B$ are invertible.
(b) Let $\lambda \in \mathbb{R}$. Show that an element $u \in G$ is a solution of $(\mathcal{P}_\lambda)$ if and only if $$(A - \lambda B) U = 0$$ Deduce that $(\mathcal{P}_\lambda)$ admits a non-zero solution if and only if $\operatorname{det}(A - \lambda B) = 0$.
(c) We define the function $\psi$ on $\mathbb{R}$ by $$\forall t \in \mathbb{R}, \psi(t) = \frac{\operatorname{det}(A - tB)}{\operatorname{det}(B)}$$ Show that this function $\psi$ is independent of the choice of basis $\mathcal{B}$.
(d) Justify why we can choose the basis $\mathcal{B}$ so that $B = I_\ell$. Deduce that $\psi$ is a polynomial function and specify its degree.
(e) Show that the polynomial $\psi$ is split over $\mathbb{R}[X]$ and that its roots are either simple or double.
(f) Show that $$\psi(X) = \frac{1}{S(w_1, T^{2m-1}(w_1)) S(w_2, T^{2m-1}(w_2))} Q_1(X) Q_2(X)$$ (justify why necessarily the denominator is non-zero). Deduce that $Q_1$ and $Q_2$ are split over $\mathbb{R}[X]$ and have simple roots.

Throughout this part, $A$ is a strictly positive matrix in $M _ { n } ( \mathbb { R } )$. By applying the previous results to the matrix ${ } ^ { t } A$, we obtain the existence of $w _ { 0 } \in \mathbb { R } ^ { n }$, whose all components are strictly positive, such that ${ } ^ { t } A w _ { 0 } = \rho ( A ) w _ { 0 }$. We set $$F = \left\{ x \in \mathbb { C } ^ { n } \mid { } ^ { t } x w _ { 0 } = 0 \right\}$$ a) Show that $F$ is a vector subspace of $\mathbb { C } ^ { n }$ stable by $\varphi _ { A }$, and that $$\mathbb { C } ^ { n } = F \oplus \mathbb { C } v _ { 0 }$$ b) Show that if $v$ is an eigenvector of $A$ associated with an eigenvalue $\mu \neq \rho ( A )$, then $v \in F$. Deduce property (iii): if $v$ is an eigenvector of $A$ whose all components are positive, then $v \in \operatorname { ker } ( A - \rho ( A ) I _ { n } )$.

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 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 $$\begin{aligned} & \omega \left( z _ { 1 } , z _ { 2 } \right) = \omega \left( y _ { 1 } , y _ { 2 } \right) = 0 \\ & \omega \left( z _ { 1 } , y _ { 1 } \right) = - \omega \left( z _ { 2 } , y _ { 2 } \right) \\ & \omega \left( z _ { 1 } , y _ { 2 } \right) = \omega \left( z _ { 2 } , y _ { 1 } \right) \end{aligned}$$

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 $+ \infty$.

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 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 ${ } ^ { t } X R _ { \theta + \phi } X > 0$.