v2007.09.13 - Convex Optimization

5.6. INJECTIVITY OF D & UNIQUE RECONSTRUCTION 3335.6.2 Inner-product form bijectivityThe Gram-form EDM operator D(G)= δ(G)1 T + 1δ(G) T − 2G (717) is aninjective map, for example, on the domain that is the subspace of symmetricmatrices having all zeros in the first row and columnS N 1∆= {G∈ S N | Ge 1 = 0}{[ ] [ 0 0T 0 0T= Y0 I 0 I] }| Y ∈ S N(1765)because it obviously has no nullspace there. Since Ge 1 = 0 ⇔ Xe 1 = 0 (719)means the first point in the list X resides at the origin, then D(G) on S N 1 ∩ S N +must be surjective onto EDM N .Substituting Θ T Θ ← −VN TDV N (782) into inner-product form EDMdefinition D(Θ) (770), it may be further decomposed: (confer (725))[ ]0D(D) =δ ( [−VN TDV ) 1 T + 1 0 δ ( −VN TDV ) ] [ ]T 0 0TN − 2N0 −VN TDV N(814)This linear operator D is another flavor of inner-product form and an injectivemap of the EDM cone onto itself. Yet when its domain is instead the entiresymmetric hollow subspace S N h = aff EDM N , D(D) becomes an injectivemap onto that same subspace. Proof follows directly from the fact: linear Dhas no nullspace [58,A.1] on S N h = aff D(EDM N )= D(aff EDM N ) (107).5.6.2.1 Inversion of D ( −VN TDV )NInjectivity of D(D) suggests inversion of (confer (722))V N (D) : S N → S N−1 ∆ = −V T NDV N (815)a linear surjective 5.27 mapping onto S N−1 having nullspace 5.28 S N⊥c ;V N (S N h ) = S N−1 (816)5.27 Surjectivity of V N (D) is demonstrated via the Gram-form EDM operator D(G):Since S N h = D(S N c ) (809), then for any Y ∈ S N−1 , −VN T †TD(VN Y V † N /2)V N = Y .5.28 N(V N ) ⊇ S N⊥c is apparent. There exists a linear mappingT(V N (D)) ∆ = V †TN V N(D)V † N = −V DV 1 2 = V(D)