Let
![$\mathscr{Q}^d$](img39.png)
be the family of all
![$d \times d$](img40.png)
positive semidefinite matrices.
We show that the form of faces of
![$\mathscr{Q}^d$](img39.png)
under geometric and algebraic definitions coincides.
Using this result, we get a classification of
![$\mathscr{Q}^d$](img39.png)
in a geometric sense.
The form of projection matrix is described more clearly, and it turns out that
![$\mathscr{Q}^d$](img39.png)
is the positive hull of all projection matrices with rank 1.