In this part, $E$ denotes a $\mathbf{C}$-vector space of dimension $n$ and $u$ denotes an endomorphism of $E$.
Assume that every vector subspace of $E$ has a complement in $E$, stable under $u$. Prove that $u$ is diagonalizable. Deduce a characterization of diagonalizable matrices in $M_{n}(\mathbf{C})$.
Hint: one may reason by contradiction and introduce a vector subspace, whose existence one will justify, of dimension $n-1$ and containing the sum of the eigenspaces of $u$.