Classifying the minimal polynomials of $n \times n$ matrices of rank $2$

Question

Classify the minimal polynomials of $n \times n$ matrices of rank $2$.

It has been hinted that we can assume that A is already in Jordan normal form - as a change of basis won't change the minimal polynomial - and we can thus examine all the possible Jordan blocks by a systematic search. But I don't understand how to utilise this procedure in answering the question.

Answer 1

Hint If $$J_{m_1}(\lambda_1) \oplus \cdots \oplus J_{m_k}(\lambda_k)$$ is the direct sum decomposition of the matrix $A$ (already in Jordan normal form) into Jordan blocks $J_{m_i}(\lambda_i)$ (respectively of size $m_i \times m_i$ and of eigenvalue $\lambda_i$), then since $$\operatorname{rank} A = \operatorname{rank}(J_{m_1}(\lambda_1) \oplus \cdots \oplus J_{m_k}(\lambda_k)) = \sum_{i = 1}^k \operatorname{rank} J_{m_i}(\lambda_i) ,$$ either:

• exactly one block is nonzero and has rank $2$, or
• exactly two blocks are nonzero, and both have rank $1$.

So, what are all of the Jordan normal form blocks of rank $1$ and $2$? What are the minimal polynomials of the resulting rank-$2$ matrices?

Answer 2

Hint: If the minimal polynomial of $A$ is $ p(x) = (x - \lambda_1)^{m_1} \cdots (x - \lambda_k)^{m_k}, $ then $m_k$ is the size of the largest Jordan block associated with $\lambda_k$.