Proving $A,B \in M_n \mathbb{C}$ are similar and finding an $X$, such that $A = X^{-1} B X$

Question

If I have two matrices $ A,B \in M_n \mathbb{C}$, what are the necessary and sufficient conditions for $A$ to be similar to $B$.

I know that they must share properties such as rank, characteristic polynomial, Frobenius normal form etc. Does the full list provided here contain all the sufficient conditions for $A$ and $B$ to be similar?

Furthermore, if you know that $A \sim B$, how does one practically calculate an $X \in GL_n \mathbb{C}$, satisfying

$$A = X^{-1} B X$$

Answer 1

Necessary and sufficient is that they have the same Jordan normal form.

In practice, you can look at $XA - B X = 0$ as a system of linear equations in the entries of $X$, and solve it. A "generic" solution will be invertible.