v2009.01.01 - Convex Optimization

392 CHAPTER 5. EUCLIDEAN DISTANCE MATRIX
injective on domain S N h because S N⊥
c ∩ S N h = 0. Revising the argument of
this inner-product form (914), we get another flavor
D ( [
−VN TDV )
N =
0
δ ( −VN TDV )
N
] [
1 T + 1 0 δ ( −VN TDV ) ] T
N
and we obtain mutual inversion of operators V N and D , for D ∈ S N h
[ 0 0
T
− 2
0 −VN TDV N
(917)
]
or
D = D ( V N (D) ) (918)
−V T NDV N = V N
(
D(−V
T
N DV N ) ) (919)
S N h = D ( V N (S N h ) ) (920)
−V T N S N h V N = V N
(
D(−V
T
N S N h V N ) ) (921)
Substituting Θ T Θ ← Φ into inner-product form EDM definition (863),
any EDM may be expressed by the new flavor
[ ]
D(Φ) =
∆ 0
1
δ(Φ)
T + 1 [ 0 δ(Φ) ] [ ] 0 0 T T
− 2
∈ EDM N
0 Φ
⇔
(922)
Φ ≽ 0
where this D is a linear surjective operator onto EDM N by definition,
injective because it has no nullspace on domain S N−1
+ . More broadly,
aff D(S N−1
+ )= D(aff S N−1
+ ) (118),
S N h = D(S N−1 )
S N−1 = V N (S N h )
(923)
demonstrably isomorphisms, and by bijectivity of this inner-product form:
such that
EDM N = D(S N−1
+ ) (924)
S N−1
+ = V N (EDM N ) (925)
N(T(V N )) = N(V) ⊇ N(V N ) ⊇ S N⊥
c
= N(V)
where the equality S N⊥
c = N(V) is known (E.7.2.0.2).