7. The Pseudoinverse
Society for Industrial and Applied Mathematics eBooks · 1995
If A is an n × n nonsingular matrix, the solution of the problem can be written as where is the (unique) inverse matrix for A. The inverse matrix is a very useful mathematical concept even though efficient and reliable contemporary methods [Forsythe and Moler (1967); Wilkinson (1965a)] for computing the solution of do not involve explicit computation of . For Problem LS there arises the question as to whether there exists some n × m matrix Z, uniquely determined by A, such that the (unique) minimum length solution of Problem LS is given by . This is indeed the case and this matrix Z is called the pseudoinverse of A. As noted above for the inverse, the pseudoinverse is a useful mathematical concept but one would not usually compute the pseudoinverse explicitly in the process of computing a solution of Problem LS.