What are the possible Jordan canonical forms for $A$?

Question

Let A be an (8 × 8) matrix over the complex numbers, and suppose that the characteristic polynomial is = $(2 − x)^8$ and the minimum polynomial is $(x − 2)^4$. What are the possible Jordan canonical forms for $A$. How would you decide which was the correct JCF?

I know that the largest Jordan block is 4x4, there's only one eigenvalue so we cannot guarantee that we have more than one L.I. eigenvector. Where do I go from here?

Answer 1

The possible Jordan forms are

1. Two 4x4 blocks
2. One 4x4 block, a 3x3 block and a 1x1 block
3. One 4x4 block and two 2x2 blocks
4. One 4x4 block, a 2x2 block and two 1x1 blocks
5. One 4x4 block and four 1x1 block

If you had the concrete matrix, you could find out which one of these it was by computing dimensions of kernels of $(A - 2I)^k$.