# Finding other generalized inverses besides the pseudoinverse?

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?

• Here's a hint: Have you thought about putting $A$ in reduced echelon form? What does that tell you? Can you write that in terms of matrix multiplication? – Ted Shifrin Apr 22 at 1:35
• Thanks @TedShifrin for the hint! Do you mean expressing the process of transforming $A$ into its reduced row echelon form as matrix multiplication? I know I can use reduced echelon form to find a square matrix's inverse. But can't see other connections between the echelon form and generalized inverse. Thanks! – Sibbs Gambling Apr 22 at 1:53
• You can get one left inverse by thinking about the equation $EA=I$, where $E$ is a product of elementary matrices. Notice that once you have one left inverse $B$, you get others $B'$ by making sure that every row of $B-B'$ is orthogonal to all the columns of $A$. – Ted Shifrin Apr 22 at 4:04

## 1 Answer

Denote by $$b_i$$ the i-th row of $$A^g$$ and by $$e_i$$ the i-th row of $$I_4$$. Then $$b_i A=e_i$$. This system of four linear equations is solveable by the rank condition for $$A$$. Following this procedure gives all possible $$A^g$$.