v2009.01.01 - Convex Optimization

390 CHAPTER 5. EUCLIDEAN DISTANCE MATRIX
because the projection of −D/2 on S N c (1874) can be 0 if and only if
D ∈ S N⊥
c ; but S N⊥
c ∩ S N h = 0 (Figure 105). Projector V on S N h is therefore
injective hence invertible. Further, −V S N h V/2 is equivalent to the geometric
center subspace S N c in the ambient space of symmetric matrices; a surjection,
S N c = V(S N ) = V ( ) ( )
S N h ⊕ S N⊥
h = V S
N
h (902)
because (65)
V ( ) ( ) (
S N h ⊇ V S
N⊥
h = V δ 2 (S N ) ) (903)
Because D(G) on S N c is injective, and aff D ( V(EDM N ) ) = D ( V(aff EDM N ) )
by property (118) of the affine hull, we find for D ∈ S N h (confer (824))
id est,
or
D(−V DV 1 2 ) = δ(−V DV 1 2 )1T + 1δ(−V DV 1 2 )T − 2(−V DV 1 2 ) (904)
D = D ( V(D) ) (905)
−V DV = V ( D(−V DV ) ) (906)
S N h = D ( V(S N h ) ) (907)
−V S N h V = V ( D(−V S N h V ) ) (908)
These operators V and D are mutual inverses.
The Gram-form D ( )
S N c (810) is equivalent to S
N
h ;
D ( S N c
) (
= D V(S
N
h ⊕ S N⊥
h ) ) = S N h + D ( V(S N⊥
h ) ) = S N h (909)
because S N h ⊇ D ( V(S N⊥
h ) ) . In summary, for the Gram-form we have the
isomorphisms [79,2] [78, p.76, p.107] [6,2.1] 5.27 [5,2] [7,18.2.1] [1,2.1]
and from the bijectivity results in5.6.1,
S N h = D(S N c ) (910)
S N c = V(S N h ) (911)
EDM N = D(S N c ∩ S N +) (912)
S N c ∩ S N + = V(EDM N ) (913)
5.27 In [6, p.6, line 20], delete sentence: Since G is also...not a singleton set.
[6, p.10, line 11] x 3 =2 (not 1).