The {2}-inverse with applications in statistics

An inverse G of a given matrix A which satisfies the property GAG = G is known as a {2}-inverse. This paper presents a three-phase inversion procedure for which the {2}-inverse is a special case. We present the geometry of {2}-inverses and show that, starting from {2}-inverses, various types of generalized inverses can be derived. Two examples of the occurrence of {2}-inverses in statistics are given: one concerning the constrained least-squares estimator, the other concerning a necessary and sufficient condition for a quadratic form of singular multivariate normal variates to follow a chi-square distribution.