The best way to compute $A^{+}$ is to use singular value decomposition. With $A=USV^{T}$ , where $U$ and $V$ (both $n\times n$) orthogonal and $S$ ($m\times n$) is diagonal with real, non-negative singular values $\sigma_{i}$, $i=1,\ldots,n$. We find

If the rank $r$ of $A$ is smaller than $n$, the inverse of $S^{T}S$ does not exist, and one uses only the first $r$ singular values; $S$ then becomes an $r\times r$ matrix and $U$,$V$ shrink accordingly. see also Linear Equations.

The term “pseudoinverse” is actually used for any operator $\operatorname{pinv}$ satisfying

for a $m\times n$ matrix $M$. Beyond this, pseudoinverses can be defined on any reasonable matrix identity.

References

• Originally from The Data Analysis Briefbook (http://rkb.home.cern.ch/rkb/titleA.html)