v2009.01.01 - Convex Optimization

404 CHAPTER 5. EUCLIDEAN DISTANCE MATRIX
λ 2 ≤ d 12 ≤ λ 1 (972)
where d 12 is the eigenvalue of submatrix (970a) and λ 1 , λ 2 are the
eigenvalues of T (970b) (963). Intertwining in (972) predicts that should
d 12 become 0, then λ 2 must go to 0 . 5.39 The eigenvalues are similarly
intertwined for submatrices (970b) and (970c);
γ 3 ≤ λ 2 ≤ γ 2 ≤ λ 1 ≤ γ 1 (973)
where γ 1 , γ 2 ,γ 3 are the eigenvalues of submatrix (970c). Intertwining
likewise predicts that should λ 2 become 0 (a possibility revealed in5.8.3.1),
then γ 3 must go to 0. Combining results so far for N = 2, 3, 4: (972) (973)
γ 3 ≤ λ 2 ≤ d 12 ≤ λ 1 ≤ γ 1 (974)
The preceding logic extends by induction through the remaining members
of the sequence (970).
5.8.3.1 Tightening the triangle inequality
Now we apply Schur complement fromA.4 to tighten the triangle inequality
from (962) in case: cardinality N = 4. We find that the gains by doing so
are modest. From (970) we identify:
[ A
] B
B T C
∆
= −V T NDV N | N←4 (975)
A ∆ = T = −V T NDV N | N←3 (976)
both positive semidefinite by assumption, where B=ν(4) (971), and
C = d 14 . Using nonstrict CC † -form (1410), C ≽0 by assumption (5.8.1)
and CC † =I . So by the positive semidefinite ordering of eigenvalues theorem
(A.3.1.0.1),
−V T NDV N | N←4 ≽ 0 ⇔ T ≽ d −1
14 ν(4)ν(4) T ⇒
{
λ1 ≥ d −1
14 ‖ν(4)‖ 2
λ 2 ≥ 0
(977)
where {d −1
14 ‖ν(4)‖ 2 , 0} are the eigenvalues of d −1
14 ν(4)ν(4) T while λ 1 , λ 2 are
the eigenvalues of T .
5.39 If d 12 were 0, eigenvalue λ 2 becomes 0 (965) because d 13 must then be equal to d 23 ;
id est, d 12 = 0 ⇔ x 1 = x 2 . (5.4)