Similar to the definition here, matrices $A$, $B$ $\in \mathbb{C}^{m\times n}$ are said to be equivalent if there exist some invertible $m\times m$ matrix $P$ and some invertible $n\times n$ matrix $Q$ such that
$$A = PBQ$$
Are there applications out there that prefer $B$ to $A$? Generally, I'm looking for applications where equivalent-matrix transforms made computation or analysis better.
The situation is perhaps clearer if one regards these matrices as linear maps $\Bbb C^n \to \Bbb C^m$. Then this equivalence literally means that up to change of coordinates in both domain and codomain, $A$ and $B$ are the same map. What happens is that if $B$ has rank $r$, there are $P$ and $Q$ such that $A$ is just the map $$\Bbb C^n \ni (x_1,\ldots,x_n) \mapsto (x_1,\ldots, x_r,0,\ldots, 0) \in \Bbb C^m,$$where we have $m-r$ zeros in the last vector. You might know the procedure done for finding $P$ and $Q$ as "reducing $B$ to row-echelon form". The row-echelon form of $B$ is $A$.