v2006.03.09 - Convex Optimization

116 CHAPTER 2. CONVEX GEOMETRY2.9.2.1 rank ρ subset of the PSD coneFor the same reason (closure), this applies more generally; for 0≤ρ≤M{A ∈ SM+ | rankA=ρ } = { A ∈ S M + | rankA≤ρ } (179)For easy reference, we give such generally nonconvex sets a name:rank ρ subset of the positive semidefinite cone. For ρ < M this subset,nonconvex for M > 1, resides on the PSD cone boundary. (We leave proofof equality an exercise.)For example,∂S M + = { A ∈ S M + | rankA=M− 1 } = { A ∈ S M + | rankA≤M− 1 } (180)In S 2 , each and every ray on the boundary of the positive semidefinite conein isomorphic R 3 corresponds to a symmetric rank-1 matrix (Figure 29), butthat does not hold in any higher dimension.2.9.2.2 Faces of PSD cone, their dimension versus rankEach and every face of the positive semidefinite cone, having dimension lessthan that of the cone, is exposed. [152,6] [133,2.3.4] Because each andevery 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 ) (132) asthe smallest face that contains A of the positive semidefinite cone S M + . ThenA , having diagonalization QΛQ T (A.5.2), is relatively interior to [17,II.12][61,31.5.3] [148,2.4]F ( S M + ∋A ) = {X ∈ S M + | N(X) ⊇ N(A)}= {X ∈ S M + | 〈Q(I − ΛΛ † )Q T , X〉 = 0}≃ S rank A+(181)which is isomorphic with the convex cone S rank A+ . Thus dimension of thesmallest face containing given matrix A isdim F ( S M + ∋A ) = rank(A)(rank(A) + 1)/2 (182)in isomorphic R M(M+1)/2 , and each and every face of S M + is isomorphic witha positive semidefinite cone having dimension the same as the face. Observe:not all dimensions are represented, and the only zero-dimensional face is theorigin. The PSD cone has no facets, for example.