Consider $A$ being a $n$ by $r$ matrix and $B$ being a $r$ by $n$ matrix. Under what conditions for $AB$ is $BA$ unique.
This question is derived from the observation that for a matrix $C = AB$ having rank $r$ and a nonzero constant multiple of an idempotent matrix, $BA$ is unique. The question is asking if above conditions necessary to gurantee uniqueness.