v2009.01.01 - Convex Optimization

5.8. EUCLIDEAN METRIC VERSUS MATRIX CRITERIA 401
T eigenvalue nonnegativity in terms of D matrix entries; videlicet,
T ≽ 0 ⇔ detT = λ 1 λ 2 ≥ 0 , d 12 ,d 13 ≥ 0 (c)
⇔
λ 2 ≥ 0
(b)
⇔
| √ d 12 − √ d 23 | ≤ √ d 13 ≤ √ d 12 + √ d 23 (a)
(966)
Triangle inequality (966a) (confer (861) (978)), in terms of three rooted
entries from D , is equivalent to metric property 4
√
d13 ≤ √ d 12 + √ d 23
√d23
≤ √ d 12 + √ d 13
√d12
≤ √ d 13 + √ d 23
(967)
for the corresponding points x 1 ,x 2 ,x 3 from some length-N list. 5.35
5.8.2.1 Comment
Given D whose dimension N equals or exceeds 3, there are N!/(3!(N − 3)!)
distinct triangle inequalities in total like (861) that must be satisfied, of which
each d ij is involved in N−2, and each point x i is in (N −1)!/(2!(N −1 − 2)!).
We have so far revealed only one of those triangle inequalities; namely, (966a)
that came from T (963). Yet we claim if −VN TDV N ≽ 0 then all triangle
inequalities will be satisfied simultaneously;
| √ d ik − √ d kj | ≤ √ d ij ≤ √ d ik + √ d kj , i