A question on characteristic polynomial and minimal polynomial

Question

I have some questions in my mind.

1. If I have two matrices A and B of order n and I say characteristic polynomial of A is same as characteristic polynomial of B then what are the things that I can predict

Like I can tell A and B have the same spectrum but may or not have same eigenvectors and eigenspaces.

Can I tell anything about their ranks? If A is diagonalisable then is B diagonalisable?

I have no idea about rank but if a is diagonalisable and has all distinct eigenvalues then b also has all distinct eigenvalues then b is diagonalisable. But what if a does not have all distinct eigenvalues?

Question 2 if A and B have same minimal polynomial and I do not know anything about characteristic polynomial

Then also I am not able to predict about ranks and diagonalisibility.

And the last question if both characteristic polynomial and minimal polynomial of A and B are same

Then what can be the inferences about rank and diagonalisibility .

Answer 1

Two matrices which have the same characteristic polynomial have the same spectrum, but not necessarily the same rank, eigenvectors or eigenspaces: the characteristic polynomial of any nilpotent matrix of rank $n$ is $X^n=0$, but $0$ is a nilpotent matrix and there exists nilpotent matrix of rank $n-1$.