v2009.01.01 - Convex Optimization

482 CHAPTER 6. CONE OF DISTANCE MATRICES
From the results inE.7.2.0.2, we know matrix product V H V is the
orthogonal projection of H ∈ S N on the geometric center subspace S N c . Thus
the projection product
P K2 H = H − P S N
+
P S N c
H (1168)
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
(1169)
⋄
Proof. For each and every H ∈ S N , projection of P S N c
H on the positive
semidefinite cone remains in the geometric center subspace
S N c = {G∈ S N | G1 = 0} (1874)
= {G∈ S N | N(G) ⊇ 1} = {G∈ S N | R(G) ⊆ N(1 T )}
= {V Y V | Y ∈ S N } ⊂ S N (1875)
(895)
That is because: eigenvectors of P S N c
H corresponding to 0 eigenvalues
span its nullspace N(P S N c
H). (A.7.3.0.1) To project P S N c
H on the positive
semidefinite cone, its negative eigenvalues are zeroed. (7.1.2) The nullspace
is thereby expanded while eigenvectors originally spanning N(P S N c
H)
remain intact. Because the geometric center subspace is invariant to
projection on the PSD cone, then the rule for projection on a convex set
in a subspace governs (E.9.5, projectors do not commute) and statement
(1169) follows directly.
From the lemma it follows
Then from (1901)
{P S N
+
P S N c
H | H ∈ S N } = {P S N
+ ∩ S N c H | H ∈ SN } (1170)
− ( S N c ∩ S N +) ∗
= {H − PS N
+
P S N c
H | H ∈ S N } (1171)
From (284) we get closure of a vector sum
K 2 = − ( )
S N c ∩ S N ∗
+ = S
N⊥
c − S N + (1172)
therefore the equality
EDM N = K 1 ∩ K 2 = S N h ∩ ( )
S N⊥
c − S N +
(1173)