The Projection Matrices with Respect to the Symmetric Nonnegative DefiniteMatrices

摘要

The projections and projection matrices are widelyused (see e.g., Basilevsky 1983; Graybill 1983; Horn andJohnson 1990; Lutkepohl 1996). The projections and projectionmatrices with respect to the symmetric positive definitematrices are often used in algebra and statistics (see e.g., Raoand Mitra 1971; Searle 1982; Harville, 1997; Abadir andMagnus 2005). Harville (1997) discussed in detail theprojections and projection matrices with respect to thesymmetric positive definite matrices and the projections andprojection matrices with respect to the symmetric nonnegativedefinite matrices. Recently, we found the projection matriceswith respect to the symmetric nonnegative definite matricesmay be able to be used in linear mixed models (see Wang2010[9-11]). However, Harville (1997) given the properties ofthe projection matrices with respect to the symmetricnonnegative definite matrices are not enough. In this paper, weextend the some properties of the projection matrices withrespect to the symmetric nonnegative definite matrices. Notethat we do not repeat the same properties with his.