Let G,P be 3x3 matrices such that G=$PEP^{-1}$. Find a matrix Q such that G=$QFQ^{-1}$

Question

I am not sure how to do this. My only thought is that if P has columns p1,p2,p3 then we would need Q to have columns p3,p2,p1. My reasoning is that since our basis will be the same eigenvectors, we just want them in a different order but if this is so, could someone explain it more rigorously.

Answer 1

Think of a permutation matrix. If $ R $ is a permutation matrix, then $ R^{-1} = ?? $. What permutation matrix $ R $ has the property that $ E = R F R^{-1} $?

I think it will then all fall in place.