v2007.09.13 - Convex Optimization

2.9. POSITIVE SEMIDEFINITE (PSD) CONE 105In 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 31), butthat does not hold in any higher dimension.2.9.2.2 Subspace tangent to open rank-ρ subsetWhen the positive semidefinite cone subset in (185) is left unclosed as inM(ρ) ∆ = { A ∈ S M + | rankA=ρ } (187)then we can specify a subspace tangent to the positive semidefinite coneat a particular member of manifold M(ρ). Specifically, the subspace R Mtangent to manifold M(ρ) at B ∈ M(ρ) [134,5, prop.1.1]R M (B) ∆ = {XB + BX T | X ∈ R M×M } ⊆ S M (188)has dimension(dim svec R M (B) = ρ M − ρ − 1 )= ρ(M − ρ) +2ρ(ρ + 1)2(189)Tangent subspace R M contains no member of the positive semidefinite coneS M + whose rank exceeds ρ .A good example of such a tangent subspace is given inE.7.2.0.2 by(1763); R M (11 T ) = S M⊥c , orthogonal complement to the geometric centersubspace. (Figure 104, p.422)2.9.2.3 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. [178,6] [157,2.3.4] 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 ) (138) asthe smallest face that contains A of the positive semidefinite cone S M + . ThenA , having ordered diagonalization QΛQ T (A.5.2), is relatively interior to[20,II.12] [77,31.5.3] [174,2.4]F ( S M + ∋A ) = {X ∈ S M + | N(X) ⊇ N(A)}= {X ∈ S M + | 〈Q(I − ΛΛ † )Q T , X〉 = 0}≃ S rank A+(190)