Finding all possible Jordan forms of an $ 8\times 8$ matrix given the minimal polynomial

Question

Find all possible Jordan forms of an $ 8\times 8$ matrix given that $$t^2(t-1)^3$$ is the minimal polynomial.

I don't really know where to start so all help would be appreciated

Answer 1

Let $J(\alpha,k)$ be the upper Jordan block with minimal polynomial $(t-\alpha)^k$. $$ \begin{pmatrix} \alpha & 1 & 0 & \ldots & \ldots\\ 0 & \alpha & 1 & 0 & \ldots \\ 0 & 0 & \alpha & \ddots & \ddots\\ \vdots & \vdots & \ddots & \ddots & \ddots \end{pmatrix} $$ So, there is - up to conjugation - one possible upper Jordan form of your matrix, in dimension 5 (see the end) $$ \begin{pmatrix} J(0,2) & 0\\ 0 & J(1,3) \end{pmatrix} $$ Now, you have to extend it to dimension 8. Calling $p_1,p_2,\cdots p_k$ (resp. $q_1,q_2,\cdots q_l$) the orders of the Jordan blocks for the eigenvalue $0$ (resp. the orders of the Jordan blocks for the eigenvalue $1$), one has $$ \sum_{i=1}^k p_i+\sum_{i=1}^l q_i=8 $$ $p_i\leq 2,\ q_i\leq 3$ and at least one of the $p_i=2$ at least one of the $q_i=3$.

Hope it helps !