= Rank orbit of a matrix under left-right multiplication
{title2=$\mathcal O_r=\{F:\operatorname{rank}F=r\}$}
= Rank orbits of a matrix under left-right multiplication
{synonym}
Two rectangular matrices are related by invertible left and right multiplication exactly when their ranks agree. The rank-$r$ orbit in $\operatorname{Mat}_{b\times a}(k)$ 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>.
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