Question about finding 2 non-similar matrices

Question

I need to find two 8x8 matrices $A,B$ with the same minimal & characteristic polynomials and same algebraic multiplicity for every eigenvalue.

I was thinking about something like $A=J_3(0), J_2(0), J_3(2)$ and $B=J_3(0), J_1(0), J_1(0), J_3(2)$.

Since their char polynomial is $(x-2)^3x^5$ with minimal polynomial $(x-2)^3x^3$ they have the same algebraic multiplicity for eigenvalues $2, 0$.

First of all, are they indeed not similar? Because these are two different jordan normal forms so are they not similar?

Also, i'm not sure but does $rk(A)\neq rk(B)$ means they aren't similar? If not, Is there any other way to determine that?

Thanks!

Answer 1

A much simpler case is $A=J_2(0)J_2(0)$ and $B=J_2(0)$. Both matrices have $\mu(x)=x^2$ for minimal polynomial and $\chi(x)=x^8$ for characteristic polynomial. Therefore their algebraic multiplicities are the same.

And indeed, two matrices with different ranks can’t be similar.

More generally, two matrices with the same characteristic polynomial that split in linear factors are similar if and only if they have the same Jordan blocks, potentially written in different orders.