A and B square matrices and A is positive semidefinite. Prove that $\displaystyle A^2B = BA^2 \Longleftrightarrow AB = BA$

The $\displaystyle \Leftarrow$ is easy:

$\displaystyle A^2B = A(AB) = A(BA) = (AB)A = (BA)A = BA^2 $

but I'm having problems with the $\displaystyle \Rightarrow$.