I'm trying to understand the following:
Claim: The fact that characteristic polynomials, minimal polynomials, and eigenspace dimensions are similarity invariants is enough to decide whether any two 4x4 complex matrices are similar.
I'm trying to go through this by considering each possible number of distinct eigenvalues.
Obviously if a matrix has 4 distinct eigenvalues, then it's diagonalizable, so only similar to other diagonalizable matrices.
If it has 3 distinct eigenvalues, then one of them has algebraic multiplicity 1 or 2. If it has 2, then it's diagonalizable, so see above. If it doesn't, then what?
I'm not sure how to proceed with the cases where the matrix has 1 or 2 distinct eigenvalues. I don't see how to use minimal or characteristic polynomials here