Prove that $T,S$ are simultaneously diagonalizable iff $TS=ST$. Definition: We say that $S,T$ are simultaneously diagonalizable if there's a basis, $B$ which composed by eigen-vectores of both $T$ and $S$

Show that $S,T$ are simultaneously diagonalizable iff $ST=TS$.

I tried both directions, but couldn't get much further. 
I'd be glad for help.
Thanks.

hint: If $S = P^{-1} D_S P$ and $T = P^{-1} D_T P$ with $D_{S,T}$ diagonal
then what are the values of $ST$ and $TS$?
