The Nature of the Lack of Uniqueness of Generalized Inverse Matrices
N. Scott Urquhart · SIAM Review · 1969
Any matrix G which satisfies $AGA = A$ is a generalized inverse of A. Unless A is square and nonsingular, G is not unique. This paper presents two characterizations of the lack of uniqueness of G: (i) in terms of a relation among elements of G and (ii) in terms of any other generalized inverse of A. The first characterization provides results concerning the rank and symmetry of G; the latter characterization provides an alternate form for the general solution to the consistent equations ${\bf Ax} = {\bf y}$, i.e., as G varies over all matrices satisfying $AGA = A$ then ${\bf x} = {\bf Gy}$ generates all solutions to ${\bf Ax} = {\bf y}$ provided ${\bf y} e 0$.