Let $W_{\lambda_i}$ be $\lambda_i$'s root subspace of linear transformation $\mathscr{A}$.
I just learned $W_{\lambda_i}$ can be represented as direct sum of some cyclic subspaces such as $W_{\lambda_i}=C_{j1}\oplus C_{j2}\oplus \cdots C_{js}$. I want to know if root space $W_{\lambda_i}$ can be represented as direct sum of some eigensubspaces? What's the relationship between cyclic subspace and eigensubspace? And I'm confused of these subspaces' geometric explanation, help.