v2010.10.26 - Convex Optimization

5.11. EDM INDEFINITENESS 471we have the Cayley-Menger form (5.7.3.0.1) of necessary and sufficientconditions for D ∈ EDM N from the literature: [184,3] 5.52 [74,3] [113,6.2](confer (910) (886))D ∈ EDM N ⇔⎧⎪⎨⎪⎩([ 0 1Tλ1 −DD ∈ S N h])≥ 0 ,ii = 1... N⎫⎪⎬⎪⎭ ⇔ {−VTN DV N ≽ 0D ∈ S N h(1107)These conditions say the Cayley-Menger form has ([ one and])only one negative0 1Teigenvalue. When D is an EDM, eigenvalues λ belong to that1 −Dparticular orthant in R N+1 having the N+1 th coordinate as sole negativecoordinate 5.53 :[ ]RN+= cone {eR 1 , e 2 , · · · e N , −e N+1 } (1108)−5.11.2.1 Cayley-Menger versus SchoenbergConnection to the Schoenberg criterion (910) is made when theCayley-Menger form is further partitioned:[ 0 1T1 −D⎡ [ ] [ ] ⎤] 0 1 1 T⎢= ⎣ 1 0 −D 1,2:N⎥⎦ (1109)[1 −D 2:N,1 ] −D 2:N,2:N[ ] 0 1Matrix D ∈ S N h is an EDM if and only if the Schur complement of1 0(A.4) in this partition is positive semidefinite; [17,1] [217,3] id est,5.52 Recall: for D ∈ S N h , −V T N DV N ≽ 0 subsumes nonnegativity property 1 (5.8.1).5.53 Empirically, all except one entry of the corresponding eigenvector have the same signwith respect to each other.