# Find the Jordan Canonical Form of a nilpotent matrix

A is a $20x20$ nilpotent matrix. $$rank(A)=11, rank(A^2)=5, rank(A^3)=2, rank(A^4)=0$$ I know that the minimal polynomial is $m_{\lambda}=\lambda^4$.

There's one eigenvalue which is $0$ (because A is nilpotent).

Because the rank is 11 the nullity is 9 so there are 9 Jordan blocks, the first one of size 4 and the rest are either all 2 or one of size 3, 6 blocks of 2 and one block of size 1.

How can I determine which is the correct form?

The$\DeclareMathOperator{null}{null}$ differences $\null[(A − λI)^j] − \null[(A − λI)^{j−1}]$ is the number of Jordan blocks associated to λ that are of size at least $j × j$.