I've been asked to enumerate all of the possible Jordan Canonical Forms for a complex matrix A. The issue is that the only information I have about A is that it's characteristic polynomial is:
$$p_A(t) = (t-3)(t-5)^3(t-7)^4$$
From here, it can be easily reasoned that the three eigenvalues for A are $\lambda = 3, \lambda = 5, \lambda = 7$, with respective algebraic multiplicities $1, 3$ and $4$.
However, I can't see any way to reason about the geometric multiplicities of the eigenvalues, hence no way of reasoning about the number of jordan blocks.
This leaves me with 36 possible choices for the J-form of A, but this seems far too many to reasonably write out - I must be missing a trick to narrow down the choices.