Write down all the possible Jordan normal forms for matrices with characteristic polynomial $(x-a)^5$. In each case, calculate the minimal polynomial and the geometric multiplicity of the eigenvalue $a$. Verify that this information determines the Jordan normal form.
I found this question in a textbook that I'm using for a test I have tomorrow. I think that I need to use the method for finding the Jordan normal form of a matrix but I can't see how to apply it and I don't have much intuition about the answer...
I'm guessing that there are 5 possibilities since the minimal polynomial can be any factor of the characteristic, but I don't know how to prove this.
Some help would be great for my test tomorrow! Thanks