 Solution

Apply the decomposition to the module on which $X$ acts as $\alpha$. On $\mathbb{C}[X]/((X-\lambda {)}^m)$ put $t=X-\lambda$ and use the ordered basisMultiplication by $X=\lambda +t$ has diagonal entries $\lambda$ and superdiagonal entries $1$, so its matrix is the Jordan block $J_m(\lambda )$. Concatenating the bases of the direct sum gives a basis of $V$ in whichThis proves existence of Jordan normal form; uniqueness of the primary cyclic decomposition also proves uniqueness of its blocks up to order.