Faces of the cone of positive semidefinite matrices

Let 2(d) be the family of all d x d positive semidefinite matrices. We show that the form of faces of 2(d) under geometric and algebraic definitions coincides. Using this result, we get a classification of 2(d) in a geometric sense. The form of projection matrix is described more clearly, and it turns out that 2(d) is the positive hull of all projection matrices with rank 1.