v2007.09.13 - Convex Optimization

358 CHAPTER 5. EUCLIDEAN DISTANCE MATRIXThese preliminary findings lead one to speculate whether any concavenondecreasing composition of individual EDM entries d ij on R + will produceanother EDM; e.g., empirical evidence suggests that given EDM D , for eachfixed α ≥ 1 [sic] the composition [log 2 (1 + d 1/αij )] is also an EDM. Figure 89illustrates that composition’s concavity in d ij together with functions from(895) and (896).5.10.1 EDM by elliptopeFor some κ∈ R + and C ∈ S N + in the elliptope E N (5.9.1.0.1), Alfakih assertsany given EDM D is expressible [7] [77,31.5]D = κ(11 T − C) ∈ EDM N (899)This expression exhibits nonlinear combination of variables κ and C . Wetherefore propose a different expression requiring redefinition of the elliptope(880) by parametrization;E n t∆= S n + ∩ {Φ∈ S n | δ(Φ)=t1} (900)where, of course, E n = E n 1 . Then any given EDM D is expressiblewhich is linear in variables t∈ R + and E∈ E N t .D = t11 T − E ∈ EDM N (901)5.11 EDM indefinitenessBy the known result inA.7.2 regarding a 0-valued entry on the maindiagonal of a symmetric positive semidefinite matrix, there can be no positivenor negative semidefinite EDM except the 0 matrix because EDM N ⊆ S N h(704) andS N h ∩ S N + = 0 (902)the origin. So when D ∈ EDM N , there can be no factorization D =A T Anor −D =A T A . [247,6.3] Hence eigenvalues of an EDM are neither allnonnegative or all nonpositive; an EDM is indefinite and possibly invertible.