v2007.11.26 - Convex Optimization

5.8. EUCLIDEAN METRIC VERSUS MATRIX CRITERIA 359T 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)(895)Triangle inequality (895a) (confer (797) (907)), in terms of three rootedentries from D , is equivalent to metric property 4√d13 ≤ √ d 12 + √ d 23√d23≤ √ d 12 + √ d 13√d12≤ √ d 13 + √ d 23(896)for the corresponding points x 1 ,x 2 ,x 3 from some length-N list. 5.345.8.2.1 CommentGiven D whose dimension N equals or exceeds 3, there are N!/(3!(N − 3)!)distinct triangle inequalities in total like (797) that must be satisfied, of whicheach 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, (895a)that came from T (892). Yet we claim if −VN TDV N ≽ 0 then all triangleinequalities will be satisfied simultaneously;| √ d ik − √ d kj | ≤ √ d ij ≤ √ d ik + √ d kj , i