I know a matrix is diagonizable iff minimal polynomial splits into linear factors. Obviously as the Jordan block would have only 1*1 blocks, containing eigenvalues. Is there any further simplification on the case where minimal and characteristic polynomials are same.
Say, characteristic polynomial and minimal polynomial of $A$ are $x^4$ , then the Jordan block would contain 0 in diagoanal entries and 1 in super diagoanal entries.and then it should not be diagonizable. Correct? Am I missing any point?