v2009.01.01 - Convex Optimization

402 CHAPTER 5. EUCLIDEAN DISTANCE MATRIX
5.8.2.1.1 Shore. The columns of Ξ r V N Ξ c hold a basis for N(1 T )
when Ξ r and Ξ c are permutation matrices. In other words, any permutation
of the rows or columns of V N leaves its range and nullspace unchanged;
id est, R(Ξ r V N Ξ c )= R(V N )= N(1 T ) (805). Hence, two distinct matrix
inequalities can be equivalent tests of the positive semidefiniteness of D on
R(V N ) ; id est, −VN TDV N ≽ 0 ⇔ −(Ξ r V N Ξ c ) T D(Ξ r V N Ξ c )≽0. By properly
choosing permutation matrices, 5.36 the leading principal submatrix T Ξ ∈ S 2
of −(Ξ r V N Ξ c ) T D(Ξ r V N Ξ c ) may be loaded with the entries of D needed to
test any particular triangle inequality (similarly to (958)-(966)). Because all
the triangle inequalities can be individually tested using a test equivalent to
the lone matrix inequality −VN TDV N ≽0, it logically follows that the lone
matrix inequality tests all those triangle inequalities simultaneously. We
conclude that −VN TDV N ≽0 is a sufficient test for the fourth property of the
Euclidean metric, triangle inequality.
5.8.2.2 Strict triangle inequality
Without exception, all the inequalities in (966) and (967) can be made
strict while their corresponding implications remain true. The then
strict inequality (966a) or (967) may be interpreted as a strict triangle
inequality under which collinear arrangement of points is not allowed.
[194,24/6, p.322] Hence by similar reasoning, −VN TDV N ≻ 0 is a sufficient
test of all the strict triangle inequalities; id est,
δ(D) = 0
D T = D
−V T N DV N ≻ 0
⎫
⎬
⎭ ⇒ √ d ij < √ d ik + √ d kj , i≠j ≠k (969)
5.8.3 −V T N DV N nesting
From (963) observe that T =−VN TDV N | N←3 . In fact, for D ∈ EDM N , the
leading principal submatrices of −VN TDV N form a nested sequence (by
inclusion) whose members are individually positive semidefinite [134] [176]
[287] and have the same form as T ; videlicet, 5.37
5.36 To individually test triangle inequality | √ d ik − √ d kj | ≤ √ d ij ≤ √ d ik + √ d kj for
particular i,k,j, set Ξ r (i,1)= Ξ r (k,2)= Ξ r (j,3)=1 and Ξ c = I .
5.37 −V DV | N←1 = 0 ∈ S 0 + (B.4.1)