v2009.01.01 - Convex Optimization

112 CHAPTER 2. CONVEX GEOMETRY
2.9.2.3 Faces of PSD cone, their dimension versus rank
Each and every face of the positive semidefinite cone, having dimension less
than that of the cone, is exposed. [212,6] [186,2.3.4] Because each and
every face of the positive semidefinite cone contains the origin (2.8.0.0.1),
each face belongs to a subspace of the same dimension.
Given positive semidefinite matrix A∈ S M + , define F ( S M + ∋A ) (151) as
the smallest face that contains A of the positive semidefinite cone S M + .
Then matrix A , having ordered diagonalization QΛQ T (A.5.2), is relatively
interior to [25,II.12] [96,31.5.3] [206,2.4] [207]
F ( S M + ∋A ) = {X ∈ S M + | N(X) ⊇ N(A)}
= {X ∈ S M + | 〈Q(I − ΛΛ † )Q T , X〉 = 0}
≃ S rank A
+
(203)
which is isomorphic with the convex cone S rank A
+ . Thus dimension of the
smallest face containing given matrix A is
dim F ( S M + ∋A ) = rank(A)(rank(A) + 1)/2 (204)
in isomorphic R M(M+1)/2 , and each and every face of S M + is isomorphic with
a positive semidefinite cone having dimension the same as the face. Observe:
not all dimensions are represented, and the only zero-dimensional face is the
origin. The positive semidefinite cone has no facets, for example.
2.9.2.3.1 Table: Rank k versus dimension of S 3 + faces
k dim F(S 3 + ∋ rank-k matrix)
0 0
boundary 1 1
2 3
interior 3 6
For the positive semidefinite cone S 2 + in isometrically isomorphic R 3
depicted in Figure 37, for example, rank-2 matrices belong to the interior
of that face having dimension 3 (the entire closed cone), rank-1 matrices