The Projection Matrices with Respect to the Symmetric Nonnegative Definite Matrices

The projections and projection matrices are widely used (see e.g., Basilevsky 1983; Graybill 1983; Horn and Johnson 1990; Lutkepohl 1996). The projections and projection matrices with respect to the symmetric positive definite matrices are often used in algebra and statistics (see e.g., Rao and Mitra 1971; Searle 1982; Harville, 1997; Abadir and Magnus 2005). Harville (1997) discussed in detail the projections and projection matrices with respect to the symmetric positive definite matrices and the projections and projection matrices with respect to the symmetric nonnegative definite matrices. Recently, we found the projection matrices with respect to the symmetric nonnegative definite matrices may be able to be used in linear mixed models (see Wang 2010[9-11]). However, Harville (1997) given the properties of the projection matrices with respect to the symmetric nonnegative definite matrices are not enough. In this paper, we extend the some properties of the projection matrices with respect to the symmetric nonnegative definite matrices. Note that we do not repeat the same properties with his.