v2009.01.01 - Convex Optimization

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).