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 the projections $p_i$, inclusions $q_i$, and $a_i = p_i a q_i$ the endomorphism of $E_i$.
Show that: $a = \sum _ { i = 1 } ^ { r } q _ { i } a _ { i } p _ { i }$.