General definition and decomposition of projectors and some applications to statistical problems

A general definition of a set of projectors for decomposing a vector as the sum of vectors belonging to disjoint subspaces not necessarily spanning the whole space is given. Such projectors are defined only over the union of the disjoint subspaces. But their extension to the whole space is of some interest in statistical problems. Explicit expressions are obtained for projectors and their extensions in terms of matrices spanning the subspaces and g-inverses. Decomposition of a projector as the sum of projectors on subspaces is obtained and applied to problems arising in correlation analysis, analysis of variance and estimation of parameters in the Gauss-Markoff model.