v2010.10.26 - Convex Optimization

534 CHAPTER 6. CONE OF DISTANCE MATRICESFrom results inE.7.2.0.2, we know matrix product V H V is theorthogonal projection of H ∈ S N on the geometric center subspace S N c . Thusthe projection productP K2 H = H − P S N+P S N cH (1263)6.8.1.1.1 Lemma. Projection on PSD cone ∩ geometric center subspace.P S N+ ∩ S N c= P S N+P S N c(1264)Proof. For each and every H ∈ S N , projection of P S N cH on the positivesemidefinite cone remains in the geometric center subspace⋄S N c = {G∈ S N | G1 = 0} (1998)= {G∈ S N | N(G) ⊇ 1} = {G∈ S N | R(G) ⊆ N(1 T )}= {V Y V | Y ∈ S N } ⊂ S N (1999)(990)That is because: eigenvectors of P S N cH corresponding to 0 eigenvaluesspan its nullspace N(P S N cH). (A.7.3.0.1) To project P S N cH on the positivesemidefinite cone, its negative eigenvalues are zeroed. (7.1.2) The nullspaceis thereby expanded while eigenvectors originally spanning N(P S N cH)remain intact. Because the geometric center subspace is invariant toprojection on the PSD cone, then the rule for projection on a convex setin a subspace governs (E.9.5, projectors do not commute) and statement(1264) follows directly. From the lemma it followsThen from (2024){P S N+P S N cH | H ∈ S N } = {P S N+ ∩ S N c H | H ∈ SN } (1265)− ( S N c ∩ S N +) ∗= {H − PS N+P S N cH | H ∈ S N } (1266)From (314) we get closure of a vector sumK 2 = − ( )S N c ∩ S N ∗+ = SN⊥c − S N + (1267)