v2007.09.17 - Convex Optimization

370 CHAPTER 5. EUCLIDEAN DISTANCE MATRIXD(X) = D(X[0 √ 2V N ]) = D(Q p [0 √ ΛQ T ]) = D([0 √ ΛQ T ]) (941)This suggests a way to find EDM D given −V T N DV N ; (confer (822))[0D =δ ( −VN TDV )N] [1 T + 1 0 δ ( −VN TDV ) ] TN[ 0 0T− 20 −VN TDV N(729)]5.12.2 0 geometric center. VAlternatively, we may perform reconstruction using instead the auxiliarymatrix V (B.4.1), corresponding to the polyhedronP − α c (942)whose geometric center has been translated to the origin. Redimensioningdiagonalization factors Q, Λ∈R N×N and unknown Q p ∈ R n×N , (845)−V DV = 2V X T XV ∆ = Q √ ΛQ T pQ p√ΛQT ∆ = QΛQ T (943)where the geometrically centered generating list constitutes (confer (939))XV = 1 √2Q p√ΛQ T ∈ R n×N= [x 1 − 1 N X1 x 2 − 1 N X1 x 3 − 1 N X1 · · · x N − 1 N X1 ] (944)where α c = 1 N X1. (5.5.1.0.1) The simplest choice for Q p is [I 0 ]∈ R r×N .Now EDM D can be uniquely made from the list found, by calculating:(709)D(X) = D(XV ) = D( 1 √2Q p√ΛQ T ) = D( √ ΛQ T ) 1 2(945)This EDM is, of course, identical to (941). Similarly to (729), from −V DVwe can find EDM D ; (confer (809))D = δ(−V DV 1 2 )1T + 1δ(−V DV 1 2 )T − 2(−V DV 1 2 ) (735)