= 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 <basis>
$$
t^{m-1},\ t^{m-2},\ldots,\ t,\ 1.
$$
Multiplication 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 which
$$
\boxed{[\alpha]=\bigoplus_{\lambda,j}J_{m_{\lambda j}}(\lambda).}
$$
This proves existence of <Jordan normal form>; uniqueness of the primary cyclic decomposition also proves uniqueness of its blocks up to order.
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