How to find invertible $P$ such that $PAP^{-1}$ is upper triangular?

Question

I know that $A$ is similar to a Jordan Normal Form matrix, but I'm struggling to find and understand a method to work out what $P$ is (NOT the JNF matrix that $A$ is similar to)

EDIT: I'm dealing with complex matrices.

Answer 1

If you already know the Jordan form $J$ of $A$, then the equation $PAP^{-1}=J$ gives $PA=JP$, which is a system of $n^2$ linear equations in the entries (as variables) of $P$, together with the condition that $\det(P)\neq 0$. This system of linear equations can be solved easily. Of course, one could also compute the generalized eigenvectors. A computation of the Jordan form usually comes together with these generalized eigenvectors (try it in Maple or Mathematica or any other CAS). The computation how to find such transformation matrix $P$ has been done already at MSE for many examples, e.g., see this question.