Showing if two matrices have the same eigenvectors, they commute
Solution 1:
You know that
$$A=PUP^{-1}$$
and
$$B=PVP^{-1}$$
for some diagonal $U,V$ (which commute). The matrix $P$ can be made the same since it's made of linearly independent eigenvectors of $A,B$ which are (can be made and ordered) the same.
$$AB=PUP^{-1}PVP^{-1} = PUVP^{-1}=\cdots$$ $$\cdots=PVUP^{-1}=PVP^{-1}PUP^{-1} = BA$$