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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.
The best way to compute  is to use singular value decomposition. With  , where  and  (both
 ) orthogonal and  (
 ) is diagonal with real, non-negative singular values  ,
 . We find
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.
References
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