v2007.09.17 - Convex Optimization

416 CHAPTER 6. EDM CONE6.7 Vectorization & projection interpretationInE.7.2.0.2 we learn: −V DV can be interpreted as orthogonal projection[4,2] of vectorized −D ∈ S N h on the subspace of geometrically centeredsymmetric matricesS N c = {G∈ S N | G1 = 0} (1766)= {G∈ S N | N(G) ⊇ 1} = {G∈ S N | R(G) ⊆ N(1 T )}= {V Y V | Y ∈ S N } ⊂ S N (1767)≡ {V N AVN T | A ∈ SN−1 }(799)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 EDMcone (704), 6.9〈zz T , −D〉 ≥ 0 ∀zz T | 11 T zz T = 0D ∈ S N h}⇔ D ∈ EDM N (1041)this is equivalent, of course, to the Schoenberg criterion−VN TDV }N ≽ 0⇔ D ∈ EDM N (728)D ∈ S N hbecause N(11 T ) = R(V N ). When D ∈ EDM N , correspondence (1041)means −z T Dz is proportional to a nonnegative coefficient of orthogonalprojection (E.6.4.2, Figure 104) of −D in isometrically isomorphicR N(N+1)/2 on the range of each and every vectorized (2.2.2.1) symmetricdyad (B.1) in the nullspace of 11 T ; id est, on each and every member ofT ∆ = { svec(zz T ) | z ∈ N(11 T )= R(V N ) } ⊂ svec ∂ S N += { svec(V N υυ T V T N ) | υ ∈ RN−1} (1042)whose dimension isdim T = N(N − 1)/2 (1043)6.9 N(11 T )= N(1 T ) and R(zz T )= R(z)