Moore-Penrose Pseudoinverse and Applications.

Abstract

An underlying theorem due to Gauss and Lengendre asserts that for an over determined system, there are solutions that minimize kAx − bk 2 which is given by the generalized in-verse of the matrix A even when A is singular or rectangular. Our objective is to prove algebraic analogs of this result for arbitrary operators on complex Hilbert spaces and its generalization for the Moore-Penrose Inverse. We employ the generalized inverse matrix of Moore-Penrose to study the existence and uniqueness of the solutions for over- and under-determined linear systems, in harmony with the least squares method.