pseudoinverse

The inverse $A^{-1}$ of a matrix $A$ exists only if $A$ is square and has full rank. In this case, $Ax=b$ has the solution $x=A^{-1}b$.

The pseudoinverse $A^{+}$ (beware, it is often denoted otherwise) is a generalization of the inverse, and exists for any $m\times n$ matrix. We assume $m>n$. If $A$ has full rank ($n$) we define:

$$A^{+}={(A^{T}A)}^{-1}{A}^T$$

and the solution of $Ax=b$ is $x=A^{+}b$.

More accurately, the above is called the Moore-Penrose pseudoinverse.

1 Calculation

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

$$A^{+}=V{(S^{T}S)}^{-1}{S}^T{U}^T$$

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.

2 Generalization

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

$$M\mathrm{pinv}(M)M=M$$

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

References

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