v2009.01.01 - Convex Optimization

416 CHAPTER 5. EUCLIDEAN DISTANCE MATRIX
These preliminary findings lead one to speculate whether any concave
nondecreasing composition of individual EDM entries d ij on R + will produce
another EDM; e.g., empirical evidence suggests that given EDM D , for
each fixed α ≥ 1 [sic] the composition [log 2 (1 + d 1/α
ij )] is also an EDM.
Figure 109 illustrates that composition’s concavity in d ij together with
functions from (997) and (998).
5.10.0.0.2 Exercise. Taxicab distance matrix as EDM.
Determine whether taxicab distance matrices (D 1 (X) in Example 3.2.0.0.2)
are all numerically equivalent to EDMs. Explain why or why not.
5.10.1 EDM by elliptope
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 [8] [96,31.5]
D = κ(11 T − C) ∈ EDM N (1001)
This expression exhibits nonlinear combination of variables κ and C . We
therefore propose a different expression requiring redefinition of the elliptope
(981) by parametrization;
E n t
∆
= S n + ∩ {Φ∈ S n | δ(Φ)=t1} (1002)
where, of course, E n = E n 1 . Then any given EDM D is expressible
which is linear in variables t∈ R + and E∈ E N t .
5.11 EDM indefiniteness
D = t11 T − E ∈ EDM N (1003)
By the known result inA.7.2 regarding a 0-valued entry on the main
diagonal of a symmetric positive semidefinite matrix, there can be no positive
nor negative semidefinite EDM except the 0 matrix because EDM N ⊆ S N h
(797) and
S N h ∩ S N + = 0 (1004)
the origin. So when D ∈ EDM N , there can be no factorization D =A T A
nor −D =A T A . [287,6.3] Hence eigenvalues of an EDM are neither all
nonnegative or all nonpositive; an EDM is indefinite and possibly invertible.