similarity invariants from Jordan form

When are two matrices 'the same operator in different clothes'? When they are similar: B = P^-1 A P for some invertible P. The Jordan form answers the classification question completely: two complex matrices are similar if and only if they have the same Jordan form (same eigenvalues with the same block sizes), up to reordering blocks.

So the COMPLETE set of similarity invariants over C is: the list of eigenvalues together with, for each eigenvalue, its partition of block sizes. Nothing else is needed and nothing can be dropped. Equivalently, the invariants are all the numbers dim ker(A - lambda*I)^k, which the rank-jump formula turns into block sizes.

Coarser invariants you already know — trace, determinant, characteristic polynomial, rank, even the minimal polynomial — are all FUNCTIONS of the Jordan data, but none of them is complete on its own. For instance the characteristic polynomial fixes eigenvalues and algebraic multiplicities but not block sizes; [lambda, 1; 0, lambda] and lambda*I share a characteristic polynomial yet are not similar.

The minimal polynomial comes closest among the classical invariants: its exponent at lambda equals the LARGEST block size for lambda. So minimal-polynomial-equals-characteristic-polynomial means exactly one block per eigenvalue (a non-derogatory / cyclic matrix). But to recover ALL block sizes you still need the full Jordan partition.