v2010.10.26 - Convex Optimization

442 CHAPTER 5. EUCLIDEAN DISTANCE MATRIXbecause the projection of −D/2 on S N c (1998) can be 0 if and only ifD ∈ S N⊥c ; but S N⊥c ∩ S N h = 0 (Figure 129). Projector V on S N h is thereforeinjective hence uniquely invertible. Further, −V S N h V/2 is equivalent to thegeometric center subspace S N c in the ambient space of symmetric matrices; asurjection,S N c = V(S N ) = V ( S N h ⊕ S N⊥h) ( )= V SNh(997)because (72)V ( ) ( ) (S N h ⊇ V SN⊥h = V δ 2 (S N ) ) (998)Because D(G) on S N c is injective, and aff D ( V(EDM N ) ) = D ( V(aff EDM N ) )by property (127) of the affine hull, we find for D ∈ S N hid est,orD(−V DV 1 2 ) = δ(−V DV 1 2 )1T + 1δ(−V DV 1 2 )T − 2(−V DV 1 2 ) (999)D = D ( V(D) ) (1000)−V DV = V ( D(−V DV ) ) (1001)S N h = D ( V(S N h ) ) (1002)−V S N h V = V ( D(−V S N h V ) ) (1003)These operators V and D are mutual inverses.The Gram-form D ( )S N c (903) 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 (1004)because S N h ⊇ D ( V(S N⊥h ) ) . In summary, for the Gram-form we have theisomorphisms [90,2] [89, p.76, p.107] [7,2.1] 5.30 [6,2] [8,18.2.1] [2,2.1]and from bijectivity results in5.6.1,S N h = D(S N c ) (1005)S N c = V(S N h ) (1006)EDM N = D(S N c ∩ S N +) (1007)S N c ∩ S N + = V(EDM N ) (1008)5.30 In [7, p.6, line 20], delete sentence: Since G is also...not a singleton set.[7, p.10, line 11] x 3 =2 (not 1).