# Geometric multiplicity of an eigenvalue, solely from the characteristic polynomial

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.

• The possibilities are given by partitions of each of the multiplicities (corresponding to possible sizes of Jordan blocks). So we have $1 + 1 + 1 = 1 + 2 = 3$ has $3$ partitions and $1 + 1 + 1 + 1 = 1 + 1 + 2 = 2 + 2 = 1 + 3 = 4$ has $5$ partitions, for a total of $15$ possible choices. Where did you get $36$? – Qiaochu Yuan Nov 14 '18 at 20:01

See from characteristics equation you can get atleast one eigenvector for each Eigenvalues. So atleast each geometric multiplicity$$\geq 1$$ now we know that algebraic multiplicity must be greater than equal to g.m. So you get upper bound. From this info you can find every possible Jordan form