The \(\{\) 2\(\}\)-inverse with applications in statistics (Q1069612)

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.