v2006.03.09 - Convex Optimization

260 CHAPTER 4. EUCLIDEAN DISTANCE MATRIXBy conservation of dimension, (A.7.2.0.1)r + dim N(VNDV T N ) = N −1 (587)r + dim N(V DV ) = N (588)For D ∈ EDM N −V T NDV N ≻ 0 ⇔ r = N −1 (589)but −V DV ⊁ 0. The general fact 4.25 (confer (493))r ≤ min{n, N −1} (590)is evident from (578) but can be visualized in the example illustrated inFigure 56. There we imagine a vector from the origin to each point in thelist. Those three vectors are linearly independent in R 3 , but affine dimensionr is 2 because the three points lie in a plane. When that plane is translatedto the origin, it becomes the only subspace of dimension r=2 that cancontain the translated triangular polyhedron.4.7.2 PrécisWe collect expressions for affine dimension: for list X ∈ R n×N and Grammatrix G∈ S N +r∆= dim(P − α) = dim P = dim conv X= dim(A − α) = dim A = dim aff X= rank(X − x 1 1 T ) = rank(X − α c 1 T )= rank Θ (517)= rankXV N = rankXV = rankXV †TN= rankX , Xe 1 = 0 or X1=0= rankVN TGV N = rankV GV = rankV † N GV N= rankG , Ge 1 = 0 (477) or G1=0 (482)= rankVN TDV N = rankV DV = rankV † N DV N = rankV N (VN TDV N)VNT= rank Λ (675)([ ]) 0 1T= N −1 − dim N1 −D[ 0 1T= rank1 −D]− 2 (598)(591)⎫⎪⎬D ∈ EDM N⎪⎭4.25 rankX ≤ min{n , N}