In this part, $a$ denotes an endomorphism of $\mathbf { C } ^ { n }$. We use the decomposition $\mathbf { C } ^ { n } = \bigoplus _ { i = 1 } ^ { r } E _ { i }$ where $E _ { i } = \operatorname { Ker } \left( a - \lambda _ { i } id _ { \mathbf { C } ^ { n } } \right) ^ { m _ { i } }$, with $a_i = p_i a q_i$ the endomorphism of $E_i$ and $\|.\|_i$ the norm on $\mathcal{L}(E_i)$. Show moreover that: $$\forall i \in \llbracket 1 ; r \rrbracket , \quad \forall t \in \mathbf { R } , \quad \left\| e ^ { t a _ { i } } \right\| _ { i } \leqslant \left| e ^ { t \lambda _ { i } } \right| \sum _ { k = 0 } ^ { m _ { i } - 1 } \frac { | t | ^ { k } } { k ! } \left\| a _ { i } - \lambda _ { i } id _ { E _ { i } } \right\| _ { i } ^ { k }$$
In this part, $a$ denotes an endomorphism of $\mathbf { C } ^ { n }$. We use the decomposition $\mathbf { C } ^ { n } = \bigoplus _ { i = 1 } ^ { r } E _ { i }$ where $E _ { i } = \operatorname { Ker } \left( a - \lambda _ { i } id _ { \mathbf { C } ^ { n } } \right) ^ { m _ { i } }$, with $a_i = p_i a q_i$ the endomorphism of $E_i$ and $\|.\|_i$ the norm on $\mathcal{L}(E_i)$.
Show moreover that:
$$\forall i \in \llbracket 1 ; r \rrbracket , \quad \forall t \in \mathbf { R } , \quad \left\| e ^ { t a _ { i } } \right\| _ { i } \leqslant \left| e ^ { t \lambda _ { i } } \right| \sum _ { k = 0 } ^ { m _ { i } - 1 } \frac { | t | ^ { k } } { k ! } \left\| a _ { i } - \lambda _ { i } id _ { E _ { i } } \right\| _ { i } ^ { k }$$