I have a $16\times 4$ matrix $A$ of rank $4$. Besides its Moore-Penrose pseudoinverse $A^+$, I'm also interested in other generalized inverses $A^g$ that satisfy $A^gA=I_4$.
Is there a way to get all of them (presumably in some analytical form with free variables)?
Are there any special inverses? By "special," I mean $A^+$ gives the solution to $Ax=y$ with minimum $\ell_2$ norm, so is there one such $A^g$ that gives the solution with minimum $\ell_0$ norm, for example?
If the answers to the two questions above are both "No," how do I find any generalized inverse other than the pseudoinverse?