THE NATURAL RIGHT AND THE NATURAL LEFT INVERSES OF RECTANGULAR MATRICES

If an $m$ by $n$ with $m < n$ matrix $A$ has a right inverse then it has infinitely many right inverses. In fact, $K (AK )^{-1}$ is a right inverse of $A$ for many $n$ by $m$ matrices $K$ of rank $m$. The natural choice for $K$ is the transpose $A'$ of $A$. Thus, we call $A'(AA')^{-1}$ the natural right inverse of $A$. It can be used (not so obviously) to solve $AX = C$ yielding the solution $X = A'(AA')^{-1}C$ which minimizes the length $||X||$. Similarly, if an $n$ by $m$ with $m < n$ matrix $B$ has a left inverse, we call $(B'B)^{-1}B'$ the natural left inverse of $B$. It can be used (in an obvious way) in an attempt "to solve" $BX =C$ yielding the best approximate solution $X =(B'B)^{-1}B'C$ which minimizes the error $|| BX||$.