v2007.09.13 - Convex Optimization

5.11. EDM INDEFINITENESS 361(confer (724) (700))⎧ ([ ])⎪⎨ 0 1Tλ≥ 0 ,D ∈ EDM N ⇔1 −Di⎪⎩D ∈ S N hi = 1... N⎫⎪⎬⎪⎭ ⇔ {−VTN DV N ≽ 0D ∈ S N h(910)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 5.48 :[ ]RN+= cone {eR 1 , e 2 , · · · e N , −e N+1 } (911)−5.11.2.1 Cayley-Menger versus SchoenbergConnection to the Schoenberg criterion (724) is made when theCayley-Menger form is further partitioned:⎡ [ ] [ ] ⎤[ ] 0 1T 0 1 1 T⎢= ⎣ 1 0 −D 1,2:N⎥⎦ (912)1 −D[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; [15,1] [158,3] id est,(confer (858))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(913)D ∈ S N h5.48 Empirically, all except one entry of the corresponding eigenvector have the same signwith respect to each other.