v2006.03.09 - Convex Optimization

264 CHAPTER 4. EUCLIDEAN DISTANCE MATRIX4.8.2 Triangle inequality property 4In light of Kreyszig’s observation [140,1.1, prob.15] that properties 2through 4 of the Euclidean metric (4.2) together imply property 1,the nonnegativity criterion (601) suggests that the matrix inequality−V T N DV N ≽ 0 might somehow take on the role of triangle inequality; id est,δ(D) = 0D T = D−V T N DV N ≽ 0⎫⎬⎭ ⇒ √ d ij ≤ √ d ik + √ d kj , i≠j ≠k (605)We now show that is indeed the case: Let T be the leading principalsubmatrix in S 2 of −VN TDV N (upper left 2×2 submatrix from (602));[T =∆1d 12 (d ]2 12+d 13 −d 23 )1(d 2 12+d 13 −d 23 ) d 13(606)Submatrix T must be positive (semi)definite whenever −VN TDV N(A.3.1.0.4,4.8.3) 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(607)where λ 1 and λ 2 are the eigenvalues of T , real due only to symmetry of T :(λ 1 = 1 d2 12 + d 13 + √ )d23 2 − 2(d 12 + d 13 )d 23 + 2(d12 2 + d13)2 ∈ R(λ 2 = 1 d2 12 + d 13 − √ )d23 2 − 2(d 12 + d 13 )d 23 + 2(d12 2 + d13)2 ∈ R(608)Nonnegativity of eigenvalue λ 1 is guaranteed by only nonnegativity of the d ijwhich in turn is guaranteed by matrix inequality (601). Inequality betweenthe eigenvalues in (607) follows from only realness of the d ij . Since λ 1always equals or exceeds λ 2 , conditions for the positive (semi)definitenessof submatrix T can be completely determined by examining λ 2 the smaller ofits two eigenvalues. A triangle inequality is made apparent when we expressT eigenvalue nonnegativity in terms of D matrix entries; videlicet,