The Projection Matrices with Respect to the Symmetric Matrices

The projections and projection matrices are widely used (see e.g., Basilevsky 1983; Gray bill 1983; Horn and Johnson 1990; Lütkepohl 1996). The projections and projection matrices with respect to the symmetric positive and nonnegative definite matrices are often used in algebra and statistics (see e.g., Rao and Mitra 1971; Searle1982; Harville, 1997; Abadir and Magnus 2005). Recently, Wang (2010) found the projection matrices with respect to the symmetric nonnegative definite matrices may be able to be used in linear mixed models. However, the Literatures given the properties of the projection matrices with respect to the symmetric nonnegative definite matrices are not enough. Wang (2010) extended the some properties of the projection matrices with respect to the symmetric nonnegative definite matrices. In this paper, we define the projection matrices with respect to the symmetric matrices, and give their properties.