v2009.01.01 - Convex Optimization

5.11. EDM INDEFINITENESS 419
we have the Cayley-Menger form (5.7.3.0.1) of necessary and sufficient
conditions for D ∈ EDM N from the literature: [158,3] 5.48 [65,3] [96,6.2]
(confer (817) (793))
D ∈ EDM N ⇔
⎧
⎪⎨
⎪⎩
([ 0 1
T
λ
1 −D
D ∈ S N h
])
≥ 0 ,
i
i = 1... N
⎫
⎪⎬
⎪⎭ ⇔ {
−V
T
N DV N ≽ 0
D ∈ S N h
(1012)
These conditions say the Cayley-Menger form has ([ one and])
only one negative
0 1
T
eigenvalue. When D is an EDM, eigenvalues λ belong to that
1 −D
particular orthant in R N+1 having the N+1 th coordinate as sole negative
coordinate 5.49 :
[ ]
R
N
+
= cone {e
R 1 , e 2 , · · · e N , −e N+1 } (1013)
−
5.11.2.1 Cayley-Menger versus Schoenberg
Connection to the Schoenberg criterion (817) is made when the
Cayley-Menger form is further partitioned:
[ 0 1
T
1 −D
⎡ [ ] [ ] ⎤
] 0 1 1 T
⎢
= ⎣ 1 0 −D 1,2:N
⎥
⎦ (1014)
[1 −D 2:N,1 ] −D 2:N,2:N
[ ] 0 1
Matrix D ∈ S N h is an EDM if and only if the Schur complement of
1 0
(A.4) in this partition is positive semidefinite; [16,1] [188,3] id est,
5.48 Recall: for D ∈ S N h , −V T N DV N ≽ 0 subsumes nonnegativity property 1 (5.8.1).
5.49 Empirically, all except one entry of the corresponding eigenvector have the same sign
with respect to each other.