Jie Wang, Lin Si: Faces of the cone of positive semidefinite matrices, 313-321

Abstract:

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

Key Words: Positive semidefinite quadratic form, cone, face, projection matrix.

2010 Mathematics Subject Classification: Primary 52A20; Secondary 11H55.