v2009.01.01 - Convex Optimization

420 CHAPTER 5. EUCLIDEAN DISTANCE MATRIX
(confer (958))
D ∈ EDM N
⇔
[ ][ ]
0 1 1
−D 2:N,2:N − [1 −D 2:N,1 ]
T
= −2VN 1 0 −D TDV N ≽ 0
1,2:N
and
(1015)
D ∈ S N h
Positive semidefiniteness of that Schur complement insures nonnegativity
(D ∈ R N×N
+ ,5.8.1), whereas complementary inertia (1417) insures existence
of that lone negative eigenvalue of the Cayley-Menger form.
Now we apply results from chapter 2 with regard to polyhedral cones and
their duals.
5.11.2.2 Ordered eigenspectra
Conditions (1012) specify eigenvalue [ membership ] to the smallest pointed
0 1
T
polyhedral spectral cone for
1 −EDM N :
K ∆ λ = {ζ ∈ R N+1 | ζ 1 ≥ ζ 2 ≥ · · · ≥ ζ N ≥ 0 ≥ ζ N+1 , 1 T ζ = 0}
[ ]
R
N
+
= K M ∩ ∩ ∂H
R −
([ ])
0 1
T
= λ
1 −EDM N
(1016)
where
∂H ∆ = {ζ ∈ R N+1 | 1 T ζ = 0} (1017)
is a hyperplane through the origin, and K M is the monotone cone
(2.13.9.4.3, implying ordered eigenspectra) which has nonempty interior but
is not pointed;
K M = {ζ ∈ R N+1 | ζ 1 ≥ ζ 2 ≥ · · · ≥ ζ N+1 } (390)
K ∗ M = { [e 1 − e 2 e 2 −e 3 · · · e N −e N+1 ]a | a ≽ 0 } ⊂ R N+1 (391)