Let m ≤ n, and A ∈ M$_{m,n}$(F) and B ∈ M$_{n,m}$(F).
$S$ = $\begin{bmatrix}AB & 0_{m,n}\\B & 0_{n,n}\end{bmatrix}$ and $T$ = $\begin{bmatrix}0_{m,m} & 0_{m,n}\\B & BA\end{bmatrix}$
If there exists a $P$ ∈ GL$_{m+n}(F)$ such that $S$ = $P$ $T$ $P^{-1}$, then $S$ and $T$ are similar.
I have tried getting the determinant and trace and characteristic polynomial but I don't think the converse works.
But how do I show that there exists a P?
If you write $SP=PT$ you can simply solve for $P\in \mathsf{GL}$ (in blocks) and obtain \begin{equation} P = \begin{pmatrix} I_m & A\\ 0 & I_n \end{pmatrix}. \end{equation}