v2009.01.01 - Convex Optimization

400 CHAPTER 5. EUCLIDEAN DISTANCE MATRIX
5.8.2 Triangle inequality property 4
In light of Kreyszig’s observation [197,1.1, prob.15] that properties 2
through 4 of the Euclidean metric (5.2) together imply nonnegativity
property 1,
2 √ d jk = √ d jk + √ d kj ≥ √ d jj = 0 , j ≠k (961)
nonnegativity criterion (957) suggests that the matrix inequality
−V T N DV N ≽ 0 might somehow take on the role of triangle inequality; id est,
δ(D) = 0
D T = D
−V T N DV N ≽ 0
⎫
⎬
⎭ ⇒ √ d ij ≤ √ d ik + √ d kj , i≠j ≠k (962)
We now show that is indeed the case: Let T be the leading principal
submatrix in S 2 of −VN TDV N (upper left 2×2 submatrix from (958));
[
T =
∆
1
d 12 (d ]
2 12+d 13 −d 23 )
1
(d 2 12+d 13 −d 23 ) d 13
(963)
Submatrix T must be positive (semi)definite whenever −VN TDV N
(A.3.1.0.4,5.8.3) Now we have,
is.
−V T N DV N ≽ 0 ⇒ T ≽ 0 ⇔ λ 1 ≥ λ 2 ≥ 0
−V T N DV N ≻ 0 ⇒ T ≻ 0 ⇔ λ 1 > λ 2 > 0
(964)
where λ 1 and λ 2 are the eigenvalues of T , real due only to symmetry of T :
(
λ 1 = 1 d
2 12 + d 13 + √ )
d23 2 − 2(d 12 + d 13 )d 23 + 2(d12 2 + d13)
2 ∈ R
(
λ 2 = 1 d
2 12 + d 13 − √ )
d23 2 − 2(d 12 + d 13 )d 23 + 2(d12 2 + d13)
2 ∈ R
(965)
Nonnegativity of eigenvalue λ 1 is guaranteed by only nonnegativity of the d ij
which in turn is guaranteed by matrix inequality (957). Inequality between
the eigenvalues in (964) follows from only realness of the d ij . Since λ 1
always equals or exceeds λ 2 , conditions for the positive (semi)definiteness of
submatrix T can be completely determined by examining λ 2 the smaller of
its two eigenvalues. A triangle inequality is made apparent when we express