v2007.09.13 - Convex Optimization

354 CHAPTER 5. EUCLIDEAN DISTANCE MATRIXBecause p ∈ P − α may be found by factoring (894), the list Θ p is foundby factoring (893). A unique EDM can be made from that list usinginner-product form definition D(Θ)| Θ=Θp (770). That EDM will be identicalto D if δ(D)=0, by injectivity of D (814).5.9.2 Necessity and sufficiencyFrom (856) we learned that matrix inequality −VN TDV N ≽ 0 is a necessarytest for D to be an EDM. In5.9.1, the connection between convex polyhedraand EDMs was pronounced by the EDM assertion; the matrix inequalitytogether with D ∈ S N h became a sufficient test when the EDM assertiondemanded that every bounded convex polyhedron have a correspondingEDM. For all N >1 (5.8.3), the matrix criteria for the existence of an EDMin (724), (879), and (700) are therefore necessary and sufficient and subsumeall the Euclidean metric properties and further requirements.5.9.3 Example revisitedNow we apply the necessary and sufficient EDM criteria (724) to an earlierproblem.5.9.3.0.1 Example. Small completion problem, III. (confer5.8.3.1.1)Continuing Example 5.3.0.0.2 pertaining to Figure 74 where N = 4,distance-square d 14 is ascertainable from the matrix inequality −VN TDV N ≽0.Because all distances in (697) are known except √ d 14 , we may simplycalculate the smallest eigenvalue of −VN TDV N over a range of d 14 as inFigure 88. We observe a unique value of d 14 satisfying (724) where theabscissa is tangent to the hypograph of the smallest eigenvalue. Sincethe smallest eigenvalue of a symmetric matrix is known to be a concavefunction (5.8.4), we calculate its second partial derivative with respect tod 14 evaluated at 2 and find −1/3. We conclude there are no other satisfyingvalues of d 14 . Further, that value of d 14 does not meet an upper or lowerbound of a triangle inequality like (867), so neither does it cause the collapseof any triangle. Because the smallest eigenvalue is 0, affine dimension r ofany point list corresponding to D cannot exceed N −2. (5.7.1.1)