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 $$F_\lambda = \operatorname{Ker}\left((A - \lambda I_n)^{m_\lambda}\right) = \operatorname{Ker}\left((u - \lambda \operatorname{Id}_E)^{m_\lambda}\right).$$ $\mathbf{20}$ ▷ Show that $\mathbf{C}^n = \bigoplus_{\lambda \in \operatorname{Sp}(A)} F_\lambda$.
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
$$F_\lambda = \operatorname{Ker}\left((A - \lambda I_n)^{m_\lambda}\right) = \operatorname{Ker}\left((u - \lambda \operatorname{Id}_E)^{m_\lambda}\right).$$
$\mathbf{20}$ ▷ Show that $\mathbf{C}^n = \bigoplus_{\lambda \in \operatorname{Sp}(A)} F_\lambda$.