Projections, generalized inverses, and quadratic forms

Abstract

The purpose of this paper is to present a unified treatment of the apparently unrelated topics, namely projections and generalized inverses, with some applications to the distribution theory of quadratic forms in Gaussian random variables. Other applications will also be mentioned. Only finite dimensional euclidean spaces are considered, but some of the theory can be extended to the infinite dimensional case using, for instance, the methods of [l]. However, the treatment then will no longer be elementary. It is customary to consider linear operators in finite dimensional spaces as matrices, and we shall do so here. For this we choose a fixed but arbitrary basis, and all considerations are related to it. Thus “projection” is synonymous with “idempotent matrix”. The properties of such matrices play a key role in what follows. The importance of projections has been noticed in the past (e.g., refs. 2-4, to name a few). However, they limited themselves to the cases of orthogonal projections, or symmetric idempotent matrices. Unless otherwise stated, the idempotent matrices are not symmetric in this paper. In Section I, several properties of such matrices are proved. It turns out that, by our methods, the results of the above authors can be simplified. In Section II, we deduce the existence (and uniqueness) of the so-called generalized inverse of a matrix [5], a computational formula [6], as well as a “spectral theorem” for such matrices using the results of Section I. Finally in Section III, we apply this theory in deriving the distributions of some quadratic forms in