v2007.09.13 - Convex Optimization

330 CHAPTER 5. EUCLIDEAN DISTANCE MATRIX∈ basis S N⊥hdim S N c = dim S N h = N(N−1)2in R N(N+1)/2dim S N⊥c= dim S N⊥h= N in R N(N+1)/2basis S N c = V {E ij }V (confer (50))S N cbasis S N⊥cS N h∈ basis S N⊥hFigure 85: Orthogonal complements in S N abstractly oriented in isometricallyisomorphic R N(N+1)/2 . Case N = 2 accurately illustrated in R 3 . Orthogonalprojection of basis for S N⊥h on S N⊥c yields another basis for S N⊥c . (Basisvectors for S N⊥c are illustrated lying in a plane orthogonal to S N c in thisdimension. Basis vectors for each ⊥ space outnumber those for its respectiveorthogonal complement; such is not the case in higher dimension.)