v2009.01.01 - Convex Optimization

5.6. INJECTIVITY OF D & UNIQUE RECONSTRUCTION 391
5.6.2 Inner-product form bijectivity
The Gram-form EDM operator D(G)= δ(G)1 T + 1δ(G) T − 2G (810) is an
injective map, for example, on the domain that is the subspace of symmetric
matrices having all zeros in the first row and column
S N 1
∆
= {G∈ S N | Ge 1 = 0}
{[ ] [ 0 0
T 0 0
T
= Y
0 I 0 I
] }
| Y ∈ S N
(1878)
because it obviously has no nullspace there. Since Ge 1 = 0 ⇔ Xe 1 = 0 (812)
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 (875) into inner-product form EDM
definition D(Θ) (863), it may be further decomposed: (confer (818))
[ ]
0
D(D) =
δ ( [
−VN TDV ) 1 T + 1 0 δ ( −VN TDV ) ] [ ]
T 0 0
T
N − 2
N
0 −VN TDV N
(914)
This linear operator D is another flavor of inner-product form and an injective
map of the EDM cone onto itself. Yet when its domain is instead the entire
symmetric hollow subspace S N h = aff EDM N , D(D) becomes an injective
map onto that same subspace. Proof follows directly from the fact: linear D
has no nullspace [75,A.1] on S N h = aff D(EDM N )= D(aff EDM N ) (118).
5.6.2.1 Inversion of D ( −VN TDV )
N
Injectivity of D(D) suggests inversion of (confer (815))
V N (D) : S N → S N−1 ∆ = −V T NDV N (915)
a linear surjective 5.28 mapping onto S N−1 having nullspace 5.29 S N⊥
c ;
V N (S N h ) = S N−1 (916)
5.28 Surjectivity of V N (D) is demonstrated via the Gram-form EDM operator D(G):
Since S N h = D(S N c ) (909), then for any Y ∈ S N−1 , −VN T †T
D(V
N Y V † N /2)V N = Y .
5.29 N(V N ) ⊇ S N⊥
c is apparent. There exists a linear mapping
T(V N (D)) ∆ = V †T
N V N(D)V † N = −V DV 1 2 = V(D)