In all that follows, $\mathbf{K = C}$. We set $E = \mathbf{C}^n$. We are given $A \in \mathcal{M}_n(\mathbf{C})$. For every eigenvalue $\lambda$ of the matrix $A$, we denote by $m_\lambda$ its multiplicity, and we introduce the vector subspace $$F_\lambda = \operatorname{Ker}\left((A - \lambda I_n)^{m_\lambda}\right).$$ $\mathbf{21}$ ▷ Deduce from question 20) the existence of three matrices $P, D$ and $N$ in $\mathcal{M}_n(\mathbf{C})$ such that $A = P(D + N)P^{-1}$, where $D$ is diagonal, $N$ is nilpotent, $DN = ND$, and $P$ is invertible.
In all that follows, $\mathbf{K = C}$. We set $E = \mathbf{C}^n$. We are given $A \in \mathcal{M}_n(\mathbf{C})$. For every eigenvalue $\lambda$ of the matrix $A$, we denote by $m_\lambda$ its multiplicity, and we introduce the vector subspace
$$F_\lambda = \operatorname{Ker}\left((A - \lambda I_n)^{m_\lambda}\right).$$
$\mathbf{21}$ ▷ Deduce from question 20) the existence of three matrices $P, D$ and $N$ in $\mathcal{M}_n(\mathbf{C})$ such that $A = P(D + N)P^{-1}$, where $D$ is diagonal, $N$ is nilpotent, $DN = ND$, and $P$ is invertible.