v2007.11.26 - Convex Optimization

350 CHAPTER 5. EUCLIDEAN DISTANCE MATRIXinjective on domain S N h because S N⊥c ∩ S N h = 0. Revising the argument ofthis inner-product form (843), we get another flavorD ( [−VN TDV )N =0δ ( −VN TDV )N] [1 T + 1 0 δ ( −VN TDV ) ] TNand we obtain mutual inversion of operators V N and D , for D ∈ S N h[ 0 0T− 20 −VN TDV N(846)]orD = D ( V N (D) ) (847)−V T NDV N = V N(D(−VTN DV N ) ) (848)S N h = D ( V N (S N h ) ) (849)−V T N S N h V N = V N(D(−VTN S N h V N ) ) (850)Substituting Θ T Θ ← Φ into inner-product form EDM definition (799),any EDM may be expressed by the new flavor[ ]D(Φ) =∆ 01δ(Φ)T + 1 [ 0 δ(Φ) ] [ ] 0 0 T T− 2∈ EDM N0 Φ⇔(851)Φ ≽ 0where this D is a linear surjective operator onto EDM N by definition,injective because it has no nullspace on domain S N−1+ . More broadly,aff D(S N−1+ )= D(aff S N−1+ ) (108),S N h = D(S N−1 )S N−1 = V N (S N h )(852)demonstrably isomorphisms, and by bijectivity of this inner-product form:such thatEDM N = D(S N−1+ ) (853)S N−1+ = V N (EDM N ) (854)N(T(V N )) = N(V) ⊇ N(V N ) ⊇ S N⊥c= N(V)where the equality S N⊥c = N(V) is known (E.7.2.0.2).