v2010.10.26 - Convex Optimization

468 CHAPTER 5. EUCLIDEAN DISTANCE MATRIX5.10.0.0.2 Exercise. Concave nondecreasing EDM-entry composition.Given EDM D = [d ij ] , empirical evidence suggests that the composition[log 2 (1 + d 1/αij )] is also an EDM for each fixed α ≥ 1 [sic] . Its concavityin d ij is illustrated in Figure 133 together with functions from (1092a) and(1093). Prove whether it holds more generally: Any concave nondecreasingcomposition of individual EDM entries d ij on R + produces another EDM.5.10.0.0.3 Exercise. Taxicab distance matrix as EDM.Determine whether taxicab distance matrices (D 1 (X) in Example 3.7.0.0.2)are all numerically equivalent to EDMs. Explain why or why not. 5.10.1 EDM by elliptope(confer (917)) For some κ∈ R + and C ∈ S N + in the elliptope E N (5.9.1.0.1),Alfakih asserts any given EDM D is expressible [9] [113,31.5]D = κ(11 T − C) ∈ EDM N (1096)This expression exhibits nonlinear combination of variables κ and C . Wetherefore propose a different expression requiring redefinition of the elliptope(1076) by scalar parametrization;E n t S n + ∩ {Φ∈ S n | δ(Φ)=t1} (1097)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 .5.11 EDM indefinitenessD = t11 T − E ∈ EDM N (1098)By known result (A.7.2) regarding a 0-valued entry on the main diagonal ofa symmetric positive semidefinite matrix, there can be no positive or negativesemidefinite EDM except the 0 matrix because EDM N ⊆ S N h (890) andS N h ∩ S N + = 0 (1099)the origin. So when D ∈ EDM N , there can be no factorization D =A T Aor −D =A T A . [331,6.3] Hence eigenvalues of an EDM are neither allnonnegative or all nonpositive; an EDM is indefinite and possibly invertible.