v2009.01.01 - Convex Optimization

468 CHAPTER 6. CONE OF DISTANCE MATRICES
6.6.0.0.1 Proof. Case r = 1 is easily proved: From the nonnegativity
development in5.8.1, extreme direction (1117), and Schoenberg criterion
(817), we need show only sufficiency; id est, prove
rankV T N DV N =1
D ∈ S N h ∩ R N×N
+
}
⇒
D ∈ EDM N
D is an extreme direction
Any symmetric matrix D satisfying the rank condition must have the form,
for z,q ∈ R N and nonzero z ∈ N(1 T ) ,
because (5.6.2.1, conferE.7.2.0.2)
D = ±(1q T + q1 T − 2zz T ) (1124)
N(V N (D)) = {1q T + q1 T | q ∈ R N } ⊆ S N (1125)
Hollowness demands q = δ(zz T ) while nonnegativity demands choice of
positive sign in (1124). Matrix D thus takes the form of an extreme
direction (1117) of the EDM cone.
The foregoing proof is not extensible in rank: An EDM
with corresponding affine dimension r has the general form, for
{z i ∈ N(1 T ), i=1... r} an independent set,
( r
)
∑ T ( r
)
∑ ∑
D = 1δ z i zi T + δ z i zi
T 1 T − 2 r z i zi T ∈ EDM N (1126)
i=1
i=1
i=1
The EDM so defined relies principally on the sum ∑ z i zi
T having positive
summand coefficients (⇔ −VN TDV N ≽0) 6.9 . Then it is easy to find a
sum incorporating negative coefficients while meeting rank, nonnegativity,
and symmetric hollowness conditions but not positive semidefiniteness on
subspace R(V N ) ; e.g., from page 418,
⎡
−V ⎣
0 1 1
1 0 5
1 5 0
6.9 (⇐) For a i ∈ R N−1 , let z i =V †T
N a i .
⎤
⎦V 1 2 = z 1z T 1 − z 2 z T 2 (1127)