v2007.09.13 - Convex Optimization

332 CHAPTER 5. EUCLIDEAN DISTANCE MATRIXbecause the projection of −D/2 on S N c (1761) can be 0 if and only ifD ∈ S N⊥c ; but S N⊥c ∩ S N h = 0 (Figure 85). Projector V on S N h is thereforeinjective hence invertible. Further, −V S N h V/2 is equivalent to the geometriccenter 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 SNh (802)because (62)V ( ) ( ) (S N h ⊇ V SN⊥h = V δ 2 (S N ) ) (803)Because D(G) on S N c is injective, and aff D ( V(EDM N ) ) = D ( V(aff EDM N ) )by property (107) of the affine hull, we find for D ∈ S N h (confer (731))id est,orD(−V DV 1 2 ) = δ(−V DV 1 2 )1T + 1δ(−V DV 1 2 )T − 2(−V DV 1 2 ) (804)D = D ( V(D) ) (805)−V DV = V ( D(−V DV ) ) (806)S N h = D ( V(S N h ) ) (807)−V S N h V = V ( D(−V S N h V ) ) (808)These operators V and D are mutual inverses.The Gram-form D ( )S N c (717) is equivalent to SNh ;D ( S N c) (= D V(SNh ⊕ S N⊥h ) ) = S N h + D ( V(S N⊥h ) ) = S N h (809)because S N h ⊇ D ( V(S N⊥h ) ) . In summary, for the Gram-form we have theisomorphisms [62,2] [61, p.76, p.107] [5,2.1] 5.26 [4,2] [6,18.2.1] [1,2.1]and from the bijectivity results in5.6.1,S N h = D(S N c ) (810)S N c = V(S N h ) (811)EDM N = D(S N c ∩ S N +) (812)S N c ∩ S N + = V(EDM N ) (813)5.26 In [5, p.6, line 20], delete sentence: Since G is also...not a singleton set.[5, p.10, line 11] x 3 =2 (not 1).