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 $\|.\|_c$ the norm on $\mathcal{L}(\mathbf{C}^n)$. Deduce the existence of a polynomial $P$ with real coefficients such that: $$\forall t \in \mathbf { R } , \quad \left\| e ^ { t a } \right\| _ { c } \leqslant P ( | t | ) \sum _ { i = 1 } ^ { r } e ^ { t \operatorname { Re } \left( \lambda _ { i } \right) }$$ where $\operatorname { Re } ( z )$ denotes the real part of a complex number $z$.
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 $\|.\|_c$ the norm on $\mathcal{L}(\mathbf{C}^n)$.
Deduce the existence of a polynomial $P$ with real coefficients such that:
$$\forall t \in \mathbf { R } , \quad \left\| e ^ { t a } \right\| _ { c } \leqslant P ( | t | ) \sum _ { i = 1 } ^ { r } e ^ { t \operatorname { Re } \left( \lambda _ { i } \right) }$$
where $\operatorname { Re } ( z )$ denotes the real part of a complex number $z$.