 Rank orbit of a matrix under left-right multiplication

Two rectangular matrices are related by invertible left and right multiplication exactly when their ranks agree. The rank-$r$ orbit in ${\mathrm{Mat}}_{b\times a}(k)$ has dimension $r(a+b-r)$: choose its image in a Grassmannian, then a surjective map to that image. These are the representation orbits of the one-arrow quiver.