v2010.10.26 - Convex Optimization

478 CHAPTER 5. EUCLIDEAN DISTANCE MATRIXD(X) = D(X[0 √ 2V N ]) = D(Q p [0 √ ΛQ T ]) = D([0 √ ΛQ T ]) (1134)This suggests a way to find EDM D given −V T N DV N (confer (1013))[0D =δ ( −VN TDV )N] [1 T + 1 0 δ ( ) ] [−VNDV T T 0 0TN − 20 −VN TDV N](1009)5.12.2 0 geometric center. VAlternatively we may perform reconstruction using auxiliary matrix V(B.4.1) and Gram matrix −V DV 1 (912) instead; to find a generating list2for polyhedronP − α c (1135)whose geometric center has been translated to the origin. Redimensioningdiagonalization factors Q, Λ∈R N×N and unknown Q p ∈ R n×N , (1036)−V DV = 2V X T XV Q √ ΛQ T pQ p√ΛQ T QΛQ T (1136)where the geometrically centered generating list constitutes (confer (1132))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 ] (1137)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: (891)D(X) = D(XV ) = D( 1 √2Q p√ΛQ T ) = D( √ ΛQ T ) 1 2(1138)This EDM is, of course, identical to (1134). Similarly to (1009), from −V DVwe can find EDM D (confer (1000))D = δ(−V DV 1 2 )1T + 1δ(−V DV 1 2 )T − 2(−V DV 1 2 ) (999)