The inverse of a matrix exists only if is square and has full rank. In this case, has the solution .
The pseudoinverse (beware, it is often denoted otherwise) is a generalization of the inverse, and exists for any matrix. We assume . If has full rank () we define:
and the solution of is .
More accurately, the above is called the Moore-Penrose pseudoinverse.
If the rank of is smaller than , the inverse of does not exist, and one uses only the first singular values; then becomes an matrix and , shrink accordingly. see also Linear Equations.
The term “pseudoinverse” is actually used for any operator satisfying
for a matrix . Beyond this, pseudoinverses can be defined on any reasonable matrix identity.
Originally from The Data Analysis Briefbook (http://rkb.home.cern.ch/rkb/titleA.htmlhttp://rkb.home.cern.ch/rkb/titleA.html)
|Date of creation||2013-03-22 12:07:21|
|Last modified on||2013-03-22 12:07:21|
|Last modified by||akrowne (2)|