v2009.01.01 - Convex Optimization

356 CHAPTER 5. EUCLIDEAN DISTANCE MATRIX
5.4.2.1 First point at origin
Assume the first point x 1 in an unknown list X resides at the origin;
Xe 1 = 0 ⇔ Ge 1 = 0 (812)
Consider the symmetric translation (I − 1e T 1 )D(G)(I − e 1 1 T ) that shifts
the first row and column of D(G) to the origin; setting Gram-form EDM
operator D(G) = D for convenience,
− ( D − (De 1 1 T + 1e T 1D) + 1e T 1De 1 1 T) 1
2 = G − (Ge 11 T + 1e T 1G) + 1e T 1Ge 1 1 T
where
e 1 ∆ =
⎡
⎢
⎣
1
0.
0
⎤
(813)
⎥
⎦ (814)
is the first vector from the standard basis. Then it follows for D ∈ S N h
G = − ( D − (De 1 1 T + 1e T 1D) ) 1
2 , x 1 = 0
= − [ 0 √ ] TD [ √ ]
2V N 0 1 2VN
2
[ ] 0 0
T
=
0 −VN TDV N
VN TGV N = −VN TDV N 1 2
∀X
(815)
where
I − e 1 1 T =
[0 √ 2V N
]
(816)
is a projector nonorthogonally projecting (E.1) on
S N 1 = {G∈ S N | Ge 1 = 0}
{ [0 √ ] T [ √ ]
= 2VN Y 0 2VN | Y ∈ S
N} (1878)
in the Euclidean sense. From (815) we get sufficiency of the first matrix
criterion for an EDM proved by Schoenberg in 1935; [270] 5.7
5.7 From (805) we know R(V N )= N(1 T ) , so (817) is the same as (793). In fact, any
matrix V in place of V N will satisfy (817) whenever R(V )= R(V N )= N(1 T ). But V N is
the matrix implicit in Schoenberg’s seminal exposition.