v2007.09.13 - Convex Optimization

2.9. POSITIVE SEMIDEFINITE (PSD) CONE 103In words, set vec X is the vector sum of the translated PSD cone(linearly mapped onto the rowspace of I ⊗ A (E)) and the nullspace ofI ⊗ A (synthesis of fact fromA.6.3 andA.7.3.0.1). Should I ⊗A have nonullspace, then vec X =(I ⊗A) −1 vec(S N + − B) which is the expected result.2.9.2 Positive semidefinite cone boundaryFor any symmetric positive semidefinite matrix A of rank ρ , there mustexist a rank ρ matrix Y such that A be expressible as an outer productin Y ; [247,6.3]A = Y Y T ∈ S M + , rankA=ρ , Y ∈ R M×ρ (182)Then the boundary of the positive semidefinite cone may be expressed∂S M + = { A ∈ S M + | rankA 1, resides on the positive semidefinite cone boundary.2.9.2.1.1 Exercise. Closure and rank ρ subset.Prove equality in (185).For example,∂S M + = { A ∈ S M + | rankA=M− 1 } = { A ∈ S M + | rankA≤M− 1 } (186)