3.II.13E

(i) Let $V$ be an $n$-dimensional vector space over $\mathbb{C}$ and let $\alpha: V \rightarrow V$ be an endomorphism. Suppose that the characteristic polynomial of $\alpha$ is $\Pi_{i=1}^{k}\left(x-\lambda_{i}\right)^{n_{i}}$, where the $\lambda_{i}$ are distinct and $n_{i}>0$ for every $i$.

Describe all possibilities for the minimal polynomial and prove that there are no further ones.

(ii) Give an example of a matrix for which both the characteristic and the minimal polynomial are $(x-1)^{3}(x-3)$.

(iii) Give an example of two matrices $A, B$ with the same rank and the same minimal and characteristic polynomials such that there is no invertible matrix $P$ with $P A P^{-1}=B$.

*Typos? Please submit corrections to this page on GitHub.*