A Simple Algorithm for Computing the Generalized Inverse of a Matrix

The generalized inverse of a matrix is important
in analysis because it provides an extension 
of the concept of an inverse which applies to all matrices.
 It also has many applications in numerical 
analysis, but it is not widely used because the existing
algorithms are fairly complicated and require 
considerable storage space.  A simple extension has
been found to the conventional orthogonalization 
method for inverting non-singular matrices, which gives
the generalized inverse with little extra effort 
and with no additional storage requirements.  The algorithm
gives the generalized inverse for any m by 
n matrix A, including the special case when m+n and A
is non-singular and the case when m>n and rank(A) 
= n.  In the first case the algorithm gives the ordinary
inverse of A.  In the second case the algorithm 
yields the ordinary least squares transformation matrix
INV(A'A)A' and has the advantage of avoiding 
the loss of significance which results in forming the product A'A explicitly.

CACM May, 1966

Rust, R.
Burrus, W. R.
Schneeberger, C.

CA660514 JB March 3, 1978  9:22 AM

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