v2006.03.09 - Convex Optimization

4.6. INJECTIVITY OF D & UNIQUE RECONSTRUCTION 253To prove injectivity of D(G) on S N c : Any matrix Y ∈ S N can bedecomposed into orthogonal components in S N ;Y = V Y V + (Y − V Y V ) (540)where V Y V ∈ S N c and Y −V Y V ∈ S N⊥c (1563). Because of translationinvariance (4.5.1.1) and linearity, D(Y −V Y V )=0 hence N(D)⊇ S N⊥c . Itremains only to showD(V Y V ) = 0 ⇔ V Y V = 0 (541)(⇔ Y = u1 T + 1u T for some u∈ R N) . D(V Y V ) will vanish whenever2V Y V = δ(V Y V )1 T + 1δ(V Y V ) T . But this implies R(1) (B.2) were asubset of R(V Y V ) , which is contradictory. Thus we haveN(D) = {Y | D(Y )=0} = {Y | V Y V = 0} = S N⊥c (542)Since G1=0 ⇔ X1=0 (481) simply means list X is geometricallycentered at the origin, and because the Gram-form EDM operator D istranslation invariant and N(D) is the translation-invariant subspace S N⊥c ,then EDM definition D(G) (538) on 4.18S N c ∩ S N + = {V Y V ≽ 0 | Y ∈ S N } ≡ {V N AV T N| A∈ S N−1+ } ⊂ S N (543)(confer5.6.1,5.7.1) must be surjective onto EDM N ; (confer (473))EDM N = { }D(G) | G ∈ S N c ∩ S N +4.6.1.1 Gram-form operator D inversionDefine the linear geometric centering operator V ; (confer (482))(544)V(D) : S N → S N ∆ = −V DV 1 2(545)[50,4.3] 4.19 This orthogonal projector V has no nullspace onS N h = aff EDM N (791)4.18 The equivalence ≡ in (543) follows from the fact: Given B = V Y V = V N AVN T ∈ SN +with only matrix A unknown, then V † †TNBV N= A and A∈ SN−1 + must be positivesemidefinite by positive semidefiniteness of B and Corollary A.3.1.0.5.4.19 Critchley cites Torgerson (1958) [225, ch.11,2] for a history and derivation of (545).