v2010.10.26 - Convex Optimization

472 CHAPTER 5. EUCLIDEAN DISTANCE MATRIX(confer (1053))D ∈ EDM N⇔[ ][ ]0 1 1−D 2:N,2:N − [1 −D 2:N,1 ]T= −2VN 1 0 −D TDV N ≽ 01,2:Nand(1110)D ∈ S N hPositive semidefiniteness of that Schur complement insures nonnegativity(D ∈ R N×N+ ,5.8.1), whereas complementary inertia (1518) insures existenceof that lone negative eigenvalue of the Cayley-Menger form.Now we apply results from chapter 2 with regard to polyhedral cones andtheir duals.5.11.2.2 Ordered eigenspectraConditions (1107) specify eigenvalue [ membership ] to the smallest pointed0 1Tpolyhedral spectral cone for1 −EDM N :K λ {ζ ∈ R N+1 | ζ 1 ≥ ζ 2 ≥ · · · ≥ ζ N ≥ 0 ≥ ζ N+1 , 1 T ζ = 0}[ ]RN+= K M ∩ ∩ ∂HR −([ ])0 1T= λ1 −EDM N(1111)where∂H {ζ ∈ R N+1 | 1 T ζ = 0} (1112)is a hyperplane through the origin, and K M is the monotone cone(2.13.9.4.3, implying ordered eigenspectra) which is full-dimensional but isnot pointed;K M = {ζ ∈ R N+1 | ζ 1 ≥ ζ 2 ≥ · · · ≥ ζ N+1 } (436)K ∗ M = { [e 1 − e 2 e 2 −e 3 · · · e N −e N+1 ]a | a ≽ 0 } ⊂ R N+1 (437)