Let $n$ and $p$ be two integers greater than or equal to 2. We fix a sequence $\left( A _ { k } \right) _ { k \in \mathbb { N } }$ of matrices of $\mathrm { GL } _ { n } ( \mathbb { C } )$ which is $p$-periodic. The sequence $\left( \Phi _ { k } \right) _ { k \in \mathbb { N } }$ of matrices of $\mathrm { GL } _ { n } ( \mathbb { C } )$ is defined by $\Phi _ { 0 } = I _ { n }$ and $\Phi _ { k + 1 } = A _ { k } \Phi _ { k }$ for all $k \in \mathbb{N}$. Prove that $\forall k \in \mathbb { N } , \Phi _ { k + p } = \Phi _ { k } \Phi _ { p }$.

Let $n$ and $p$ be two integers greater than or equal to 2. We fix a sequence $\left( A _ { k } \right) _ { k \in \mathbb { N } }$ of matrices of $\mathrm { GL } _ { n } ( \mathbb { C } )$ which is $p$-periodic. The sequence $\left( \Phi _ { k } \right) _ { k \in \mathbb { N } }$ is defined by $\Phi _ { 0 } = I _ { n }$ and $\Phi _ { k + 1 } = A _ { k } \Phi _ { k }$. We denote by $B$ a matrix belonging to $\mathrm { GL } _ { n } ( \mathbb { C } )$ satisfying $B ^ { p } = \Phi _ { p }$. Prove that there exists a unique sequence $\left( P _ { k } \right) _ { k \in \mathbb { N } } \in \left( \mathrm { GL } _ { n } ( \mathbb { C } ) \right) ^ { \mathbb { N } }$, periodic of period $p$, such that $$\forall k \in \mathbb { N } , \quad \Phi _ { k } = P _ { k } B ^ { k }$$

Let $n$ and $p$ be two integers greater than or equal to 2. We fix a sequence $\left( A _ { k } \right) _ { k \in \mathbb { N } }$ of matrices of $\mathrm { GL } _ { n } ( \mathbb { C } )$ which is $p$-periodic. The sequence $\left( \Phi _ { k } \right) _ { k \in \mathbb { N } }$ is defined by $\Phi _ { 0 } = I _ { n }$ and $\Phi _ { k + 1 } = A _ { k } \Phi _ { k }$. We denote by $B$ a matrix in $\mathrm { GL } _ { n } ( \mathbb { C } )$ satisfying $B ^ { p } = \Phi _ { p }$, and $\left( P _ { k } \right) _ { k \in \mathbb { N } }$ the unique $p$-periodic sequence in $\left( \mathrm { GL } _ { n } ( \mathbb { C } ) \right) ^ { \mathbb { N } }$ such that $\Phi _ { k } = P _ { k } B ^ { k }$ for all $k \in \mathbb{N}$. Let $\left( Y _ { k } \right) _ { k \in \mathbb { N } }$ be a solution of (III.1). Justify the existence of $M = \max _ { k \in \mathbb { N } } \left\| P _ { k } \right\| _ { 0 }$. Show that for all $k \in \mathbb { N } , \left\| \Phi _ { k } \right\| _ { 0 } \leqslant n M \left\| B ^ { k } \right\| _ { 0 }$.

Let $n$ and $p$ be two integers greater than or equal to 2. We fix a sequence $\left( A _ { k } \right) _ { k \in \mathbb { N } }$ of matrices of $\mathrm { GL } _ { n } ( \mathbb { C } )$ which is $p$-periodic. The sequence $\left( \Phi _ { k } \right) _ { k \in \mathbb { N } }$ is defined by $\Phi _ { 0 } = I _ { n }$ and $\Phi _ { k + 1 } = A _ { k } \Phi _ { k }$. We denote by $B$ a matrix in $\mathrm { GL } _ { n } ( \mathbb { C } )$ satisfying $B ^ { p } = \Phi _ { p }$, and $\left( P _ { k } \right) _ { k \in \mathbb { N } }$ the unique $p$-periodic sequence such that $\Phi _ { k } = P _ { k } B ^ { k }$. Let $\left( Y _ { k } \right) _ { k \in \mathbb { N } }$ be a solution of (III.1).
a) Prove that if $\lim _ { k \rightarrow + \infty } \left\| B ^ { k } \right\| _ { 0 } = 0$, then $\lim _ { k \rightarrow + \infty } \left\| Y _ { k } \right\| _ { \infty } = 0$.
b) Prove that if the sequence $\left( \left\| B ^ { k } \right\| _ { 0 } \right) _ { k \in \mathbb { N } }$ is bounded, then the sequence $\left( \left\| Y _ { k } \right\| _ { \infty } \right) _ { k \in \mathbb { N } }$ is also bounded.

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 fix a natural number $p$ greater than or equal to 2. We denote $\mathcal { V } _ { p } = \left\{ \mathrm { e } ^ { \frac { 2 \mathrm { i } k \pi } { p } } ; k \in \llbracket 1 , p - 1 \rrbracket \right\}$, the set of $p$-th roots of unity different from 1. Let $A = \left( a _ { i , j } \right)$ be a matrix in $\mathcal { M } _ { n } ( \mathbb { C } )$ that is upper triangular and invertible. Let $\lambda$ be a non-zero complex number. Assume that, for all $i \in \llbracket 1 , n \rrbracket , \frac { a _ { i , i } } { \lambda } \notin \mathcal { V } _ { p }$. Prove that the matrix $\sum _ { j = 0 } ^ { p - 1 } \lambda ^ { p - 1 - j } A ^ { j }$ is invertible.

We fix a natural number $p$ greater than or equal to 2. Prove that every invertible matrix in $\mathcal { M } _ { n } ( \mathbb { C } )$ admits at least one $p$-th root.
One may use the fact that every upper triangular and invertible matrix admits at least one upper triangular $p$-th root (as established in V.B.3).

a) For any matrix $M \in M _ { n } ( \mathbb { C } )$ and any real number $C > 0$, show the equivalence $$\| M \| \leqslant C \Longleftrightarrow \forall x \in \mathbb { C } ^ { n } : \| M x \| _ { 1 } \leqslant C \| x \| _ { 1 } .$$ b) Show that the map $M \longmapsto \| M \|$ is a norm on $M _ { n } ( \mathbb { C } )$.

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

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 $T^2$.

Show that for $A , B \in M _ { n } ( \mathbb { C } ) , \| A B \| \leqslant \| A \| \| B \|$.

Show that $\omega ( x , x ) = 0$ for all $\omega \in \mathrm { A } ( E )$ and for all $x \in E$.

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 $A \in M _ { n } ( \mathbb { C } )$. We denote by $a _ { i , j }$ the coefficient of $A$ with row index $i$ and column index $j$. Show that $$\| A \| = \max _ { 1 \leqslant j \leqslant n } \left( \sum _ { i = 1 } ^ { n } \left| a _ { i , j } \right| \right)$$

Let $\omega \in \mathrm { A } ( E )$ and $\mathcal { B } = \left( b _ { 1 } , \ldots , b _ { n } \right)$ a basis of $E$.
(a) Show that there exists a unique matrix $M \in \mathcal { M } _ { n } ( \mathbb { R } )$, whose coefficients we shall specify, such that for all $( x , y ) \in E ^ { 2 } , \omega ( x , y ) = { } ^ { t } X M Y$ where $X , Y \in \mathbb { R } ^ { n }$ are the column matrices representing respectively $x$ and $y$ in the basis $\mathcal { B }$: $$X = \left( \begin{array} { c } x _ { 1 } \\ \vdots \\ x _ { n } \end{array} \right) , \quad Y = \left( \begin{array} { c } y _ { 1 } \\ \vdots \\ y _ { n } \end{array} \right) , \quad \begin{aligned} & x = x _ { 1 } b _ { 1 } + \cdots + x _ { n } b _ { n } \\ & y = y _ { 1 } b _ { 1 } + \cdots + y _ { n } b _ { n } . \end{aligned}$$ We then denote $M = \operatorname { Mat } _ { \mathcal { B } } ( \omega )$.
(b) Show that $M$ is antisymmetric, that is, ${ } ^ { t } M = - M$.
(c) Show that the vector space $\mathrm { A } ( E )$ is of dimension 1 when $E$ is of dimension 2.
(d) Show the equivalence between the three following statements.
$\left( \mathcal { E } _ { 1 } \right) : \quad \omega$ is a symplectic form on $E$.
$\left( \mathcal { E } _ { 2 } \right) : \quad$ For all $x \in E \backslash \{ 0 \}$, there exists $y \in E$ such that $\omega ( x , y ) \neq 0$.
$\left( \mathcal { E } _ { 3 } \right) : \quad \operatorname { Mat } _ { \mathcal { B } } ( \omega )$ is invertible.

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 $$\operatorname{dim}(\operatorname{Im}(T^k)) = 2m+1-k, \quad \operatorname{dim}(\operatorname{ker}(T^k)) = k$$

We say that a sequence $\left( A ^ { ( k ) } \right) _ { k \in \mathbb { N } }$ of matrices in $M _ { n } ( \mathbb { C } )$ converges to a matrix $B \in M _ { n } ( \mathbb { C } )$ when $$\forall i \in \llbracket 1 , n \rrbracket , \forall j \in \llbracket 1 , n \rrbracket , \lim _ { k \rightarrow + \infty } \left( a _ { i , j } \right) ^ { ( k ) } = b _ { i , j }$$ Show that the sequence $( A ^ { ( k ) } )$ converges to $B$ if and only if $\lim _ { k \rightarrow + \infty } \left\| A ^ { ( k ) } - B \right\| = 0$.

Show that, if there exists a symplectic form on $E$, then $E$ is of even dimension.

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 $0 \leq k \leq 2m+1$.

We consider in this question a matrix $A \in M _ { n } ( \mathbb { C } )$ that is upper triangular, $$A = \left( \begin{array} { c c c c c } a _ { 1,1 } & a _ { 1,2 } & \ldots & \ldots & a _ { 1 , n } \\ 0 & a _ { 2,2 } & \ldots & \ldots & a _ { 2 , n } \\ \vdots & \ddots & \ddots & & \vdots \\ \vdots & & \ddots & \ddots & \vdots \\ 0 & \ldots & \ldots & 0 & a _ { n , n } \end{array} \right)$$ We assume that $$\forall i \in \llbracket 1 , n \rrbracket , \left| a _ { i , i } \right| < 1$$ For any real $b > 0$, we set $P _ { b } = \operatorname { diag } \left( 1 , b , b ^ { 2 } , \ldots , b ^ { n - 1 } \right) \in M _ { n } ( \mathbb { R } )$. a) Compute $P _ { b } ^ { - 1 } A P _ { b }$. What happens when $b$ tends to 0? b) Show that there exists $b > 0$ such that $$\left\| P _ { b } ^ { - 1 } A P _ { b } \right\| < 1$$ c) Deduce that the sequence $\left( A ^ { k } \right) _ { k \in \mathbb { N } ^ { * } }$ converges to 0.

Show that the map $\omega _ { 0 }$ defined by $$\begin{array} { r l c c } \omega _ { 0 } : & \mathbb { R } ^ { n } \times \mathbb { R } ^ { n } & \rightarrow & \mathbb { R } \\ ( X , Y ) & \mapsto & { } ^ { t } X J _ { n } Y \end{array}$$ is a symplectic form on $\mathbb { R } ^ { n }$.

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 $T^{2m+1-k}(z) \neq 0_E$.

Determine the spectral radius of the following matrices $$\left( \begin{array} { l l } 0 & 0 \\ 0 & 1 \end{array} \right) , \quad \left( \begin{array} { l l } 0 & 0 \\ 1 & 0 \end{array} \right) , \quad \left( \begin{array} { l l } 1 & 0 \\ 0 & 0 \end{array} \right) , \quad \left( \begin{array} { c c } 0 & - 1 \\ 2 & 0 \end{array} \right) , \quad \left( \begin{array} { l l } 3 & 2 \\ 1 & 2 \end{array} \right)$$

We fix a symplectic form $\omega$ on $E$. The purpose of questions 6 to 9 is to show that there exists a basis $\mathcal { B }$ of $E$ such that $\operatorname { Mat_{\mathcal {B}} } ( \omega ) = J _ { n }$.
Treat the case where $E$ is of dimension 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 any real number $\alpha$, the endomorphism $\operatorname{Id}_E + \alpha T^2$ is bijective and that $$\left(\operatorname{Id}_E + \alpha T^2\right)^{-1} = \sum_{k=0}^{m} (-1)^k \alpha^k T^{2k}$$ where $\left(\operatorname{Id}_E + \alpha T^2\right)^{-1}$ denotes the inverse endomorphism of $\operatorname{Id}_E + \alpha T^2$.