v2009.01.01 - Convex Optimization
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v2009.01.01 - Convex Optimization

v2009.01.01 - Convex Optimization v2009.01.01 - Convex Optimization

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388 CHAPTER 5. EUCLIDEAN DISTANCE MATRIX ∈ basis S N⊥ h dim S N c = dim S N h = N(N−1) 2 in R N(N+1)/2 dim S N⊥ c = dim S N⊥ h = N in R N(N+1)/2 basis S N c = V {E ij }V (confer (52)) S N c basis S N⊥ c S N h ∈ basis S N⊥ h Figure 105: Orthogonal complements in S N abstractly oriented in isometrically isomorphic R N(N+1)/2 . Case N = 2 accurately illustrated in R 3 . Orthogonal projection of basis for S N⊥ h on S N⊥ c yields another basis for S N⊥ c . (Basis vectors for S N⊥ c are illustrated lying in a plane orthogonal to S N c in this dimension. Basis vectors for each ⊥ space outnumber those for its respective orthogonal complement; such is not the case in higher dimension.)

5.6. INJECTIVITY OF D & UNIQUE RECONSTRUCTION 389 To prove injectivity of D(G) on S N c : Any matrix Y ∈ S N can be decomposed into orthogonal components in S N ; Y = V Y V + (Y − V Y V ) (896) where V Y V ∈ S N c and Y −V Y V ∈ S N⊥ c (1876). Because of translation invariance (5.5.1.1) and linearity, D(Y −V Y V )=0 hence N(D)⊇ S N⊥ c . It remains only to show D(V Y V ) = 0 ⇔ V Y V = 0 (897) ( ⇔ Y = u1 T + 1u T for some u∈ R N) . D(V Y V ) will vanish whenever 2V Y V = δ(V Y V )1 T + 1δ(V Y V ) T . But this implies R(1) (B.2) were a subset of R(V Y V ) , which is contradictory. Thus we have N(D) = {Y | D(Y )=0} = {Y | V Y V = 0} = S N⊥ c (898) Since G1=0 ⇔ X1=0 (819) simply means list X is geometrically centered at the origin, and because the Gram-form EDM operator D is translation invariant and N(D) is the translation-invariant subspace S N⊥ c , then EDM definition D(G) (894) on 5.25 (confer6.6.1,6.7.1,A.7.4.0.1) S N c ∩ S N + = {V Y V ≽ 0 | Y ∈ S N } ≡ {V N AV T N must be surjective onto EDM N ; (confer (811)) EDM N = { D(G) | G ∈ S N c ∩ S N + 5.6.1.1 Gram-form operator D inversion | A∈ S N−1 + } ⊂ S N (899) Define the linear geometric centering operator V ; (confer (820)) } (900) V(D) : S N → S N ∆ = −V DV 1 2 (901) [78,4.3] 5.26 This orthogonal projector V has no nullspace on S N h = aff EDM N (1154) 5.25 Equivalence ≡ in (899) follows from the fact: Given B = V Y V = V N AVN T ∈ SN + with only matrix A unknown, then V † †T NBV N = A and A∈ SN−1 + must be positive semidefinite by positive semidefiniteness of B and Corollary A.3.1.0.5. 5.26 Critchley cites Torgerson (1958) [302, ch.11,2] for a history and derivation of (901).

388 CHAPTER 5. EUCLIDEAN DISTANCE MATRIX<br />

∈ basis S N⊥<br />

h<br />

dim S N c = dim S N h = N(N−1)<br />

2<br />

in R N(N+1)/2<br />

dim S N⊥<br />

c<br />

= dim S N⊥<br />

h<br />

= N in R N(N+1)/2<br />

basis S N c = V {E ij }V (confer (52))<br />

S N c<br />

basis S N⊥<br />

c<br />

S N h<br />

∈ basis S N⊥<br />

h<br />

Figure 105: Orthogonal complements in S N abstractly oriented in<br />

isometrically isomorphic R N(N+1)/2 . Case N = 2 accurately illustrated in R 3 .<br />

Orthogonal projection of basis for S N⊥<br />

h on S N⊥<br />

c yields another basis for S N⊥<br />

c .<br />

(Basis vectors for S N⊥<br />

c are illustrated lying in a plane orthogonal to S N c in this<br />

dimension. Basis vectors for each ⊥ space outnumber those for its respective<br />

orthogonal complement; such is not the case in higher dimension.)

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