Further contributions to the theory of generalized inverse of matrices and its applications

Abstract

This is a sequel to an earlier paper by the authors on the same subject presented at the Sixth Berkeley Symposium. In the previous paper, the authors have discussed there basic types of g-inverses-the minimum norm g-inverse, the least squares g-inverse and the minimum norm least squares g-inverse. In the paper these concepts are extended to more general situations involving semi norms in place of norms used earlier. It shown that a matrix is uniquely determined by its class of g-inverses. Further the subclass of g-inverses with a specified rank is characterized. Partial isometrics are discussed in a general set-up with reference to a pair linear spaces furnished with arbitrary quadratic norms. A unified theory of linear estimation is presented using the expression for a minimum semi norm inverse.