In this part, $a$ denotes an endomorphism of $\mathbf { C } ^ { n }$. Show that there exist a non-zero natural number $r$, distinct complex numbers $\lambda _ { 1 } , \lambda _ { 2 } , \ldots$, $\lambda _ { r }$, and non-zero natural numbers $m _ { 1 } , m _ { 2 } , \ldots , m _ { r }$, such that: $$\mathbf { C } ^ { n } = \bigoplus _ { i = 1 } ^ { r } E _ { i }$$ where for $i \in \llbracket 1 ; r \rrbracket , E _ { i } = \operatorname { Ker } \left( a - \lambda _ { i } id _ { \mathbf { C } ^ { n } } \right) ^ { m _ { i } }$.
In this part, $a$ denotes an endomorphism of $\mathbf { C } ^ { n }$.
Show that there exist a non-zero natural number $r$, distinct complex numbers $\lambda _ { 1 } , \lambda _ { 2 } , \ldots$, $\lambda _ { r }$, and non-zero natural numbers $m _ { 1 } , m _ { 2 } , \ldots , m _ { r }$, such that:
$$\mathbf { C } ^ { n } = \bigoplus _ { i = 1 } ^ { r } E _ { i }$$
where for $i \in \llbracket 1 ; r \rrbracket , E _ { i } = \operatorname { Ker } \left( a - \lambda _ { i } id _ { \mathbf { C } ^ { n } } \right) ^ { m _ { i } }$.