15 Generalized inverse of matrices and applications to linear models

Abstract

This chapter focuses on the generalized inverse of matrices and applications to linear models and considers only those inverses that have been found useful in the study of linear models. It focuses on the presentation of various g-inverses and on the description of their uses rather than on formal mathematics. Only those theorems are stated in the chapter that aid in the understanding of these inverses. The chapter discusses reflexive generalized inverse, minimum seminorm g-inverse, semileast squares inverse, minimum seminorm semileast square inverse, optimal inverse, constrained inverse, generalized inverse of partitioned matrices, the intersection of vector subspaces, statistical analysis of a linear model, linear estimation in a general Gauss-Markov model, the tests of linear hypotheses, Bayes linear and minimax linear estimators, best linear minimum bias estimator (BLIMBE), and Hoerl–Kennard and James–Stein estimators.