$A$ is a $10X10$ matrix.
So from this I know that:
- All the eigenvalues are $0$ (so i'll just talk about the blocks's sizes)
- The minimal polynomial is $\lambda^3$, so the largest Jordan block is of size 3
- There are 2 blocks of at least size 3 (because of the nullity difference between the ranks) $\implies$ There are 3 blocks of size 3.
So I think the possible JCFs are:
- Two blocks of size 3, two blocks of size 2
- Two blocks of size 3, one block of size 2, two blocks of size 1
- Two blocks of size 3, four blocks of size 1
But I'm not sure how to find the dimension of the eigenspaces for $A^2,A^3$. Is it just the rank of the matrix itself?