Let $A$ be a tall matrix, $n\times m$ with $n>m$. Suppose it has full rank. It is a fact that $A$ will have infinitely many left inverses. I would like to know if there are interesting conditions that we can impose on $B$ to make it unique.
For instance the condition that $AB$ be symmetric seems fairly strong, and is at least satisfied by the pseudo-inverse. Does this in fact completely determine $B$? I haven't been able to find a counterexample.
I'm also interested in other interesting ideas along these lines, though this is more open ended and less well-defined a question.