Finding all possible Jordan Normal Forms

Question

I need help with finding all Jordan Normal Forms with following infos: $F$ is an endomorphism of $V$, $\:\dim(V) = 8$, $\:\operatorname{rank}(F) = 5$, $\:\operatorname{rank}(F^2) = 4$, $\:\operatorname{rank}(F^3) = 3$, $\dim \ker(F-\operatorname{id}) = 2$, $\:\dim \ker(F-\operatorname{id})^2 = 3$. All we learned in class is finding Jordan Normal Forms with characteristic and minimal polynomials.

Answer 1

Let me introduce the following notations for solving this problem: $m_k$=dimension of (F-aI)^k where I is the Identity matrix and "a" is an eigen value of F. Also let $b_k$=Number of k blocks (k by k) for an eigen value "a".

For the eigen value "0" we observe the following from the given data. Since rank(F)=5 we have nullity(F)=3 and hence "0" has geometric multiplicity 3. Also we have $m_0$=0 (always for any eigen value), $m_1$=3; $m_2$=4 and $m_3$=5. Hence $b_1$=number of one blocks for eigen value "0" = 2$m_1$-$m_2$-$m_0$=6-4-0=2. $b_2$=2$m_2$-$m_3$-$m_1$=8-5-3=0. So eigen value "0" has no 2 blocks but since its gm is 3 it must have a 3 or higher block. Repeating the calculations for eigen value "1" we conclude that it has one 1 block and at least one higher block. Putting all together we have Eigen value "0" repeated 5 times (AM=5) with two 1 blocks and one 3 block and Eigen value "1" with AM=3 with one 1 block and one 2 block and hence the JNF is uniquely determined by the given data as [0,0,(0,0,0),1,(1,1)].