v2010.10.26 - Convex Optimization

122 CHAPTER 2. CONVEX GEOMETRY2.9.2.3 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. [246,6] [215,2.3.4] Because each andevery face of the positive semidefinite cone contains the origin (2.8.0.0.1),each face belongs to a subspace of dimension the same as the face.Define F(S M + ∋A) (171) as the smallest face, that contains a givenpositive semidefinite matrix A , of positive semidefinite cone S M + . Thenmatrix A , having ordered diagonalization A = QΛQ T ∈ S M + (A.5.1), isrelatively interior to 2.39 [26,II.12] [113,31.5.3] [239,2.4] [240]F ( S M + ∋A ) = {X ∈ S M + | N(X) ⊇ N(A)}= {X ∈ S M + | 〈Q(I − ΛΛ † )Q T , X〉 = 0}= {QΛΛ † ΨΛΛ † Q T | Ψ ∈ S M + }= QΛΛ † S M + ΛΛ † Q T≃ S rank A+(221)which is isomorphic with convex cone S rank A+ ; e.g., Q S M + Q T = S M + . Thelarger the nullspace of A , the smaller the face. (140) Thus dimension of thesmallest face that contains given matrix A isdim F ( S M + ∋A ) = rank(A)(rank(A) + 1)/2 (222)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 positive semidefinite cone has no facets, for example.2.9.2.3.1 Table: Rank k versus dimension of S 3 + facesk dim F(S 3 + ∋ rank-k matrix)0 0boundary ≤1 1≤2 3interior ≤3 62.39 For X ∈ S M + , A=QΛQ T ∈ S M + , show N(X) ⊇ N(A) ⇔ 〈Q(I − ΛΛ † )Q T , X 〉 = 0.Given 〈Q(I − ΛΛ † )Q T , X 〉 = 0 ⇔ R(X)⊥ N(A). (A.7.4)(⇒) Assume N(X) ⊇ N(A), then R(X)⊥ N(X) ⊇ N(A).(⇐) Assume R(X)⊥ N(A), then X Q(I − ΛΛ † )Q T = 0 ⇒ N(X) ⊇ N(A).