v2009.01.01 - Convex Optimization

472 CHAPTER 6. CONE OF DISTANCE MATRICES
6.7 Vectorization & projection interpretation
InE.7.2.0.2 we learn: −V DV can be interpreted as orthogonal projection
[5,2] of vectorized −D ∈ S N h on the subspace of geometrically centered
symmetric matrices
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)
≡ {V N AVN T | A ∈ SN−1 }
(895)
because elementary auxiliary matrix V is an orthogonal projector (B.4.1).
Yet there is another useful projection interpretation:
Revising the fundamental matrix criterion for membership to the EDM
cone (793), 6.10
〈zz T , −D〉 ≥ 0 ∀zz T | 11 T zz T = 0
D ∈ S N h
}
⇔ D ∈ EDM N (1140)
this is equivalent, of course, to the Schoenberg criterion
−VN TDV }
N ≽ 0
⇔ D ∈ EDM N (817)
D ∈ S N h
because N(11 T ) = R(V N ). When D ∈ EDM N , correspondence (1140)
means −z T Dz is proportional to a nonnegative coefficient of orthogonal
projection (E.6.4.2, Figure 123) of −D in isometrically isomorphic
R N(N+1)/2 on the range of each and every vectorized (2.2.2.1) symmetric
dyad (B.1) in the nullspace of 11 T ; id est, on each and every member of
T ∆ = { svec(zz T ) | z ∈ N(11 T )= R(V N ) } ⊂ svec ∂ S N +
= { svec(V N υυ T V T N ) | υ ∈ RN−1} (1141)
whose dimension is
dim T = N(N − 1)/2 (1142)
6.10 N(11 T )= N(1 T ) and R(zz T )= R(z)