v2010.10.26 - Convex Optimization
v2010.10.26 - Convex Optimization v2010.10.26 - Convex Optimization
448 CHAPTER 5. EUCLIDEAN DISTANCE MATRIXBy conservation of dimension, (A.7.3.0.1)r + dim N(VNDV T N ) = N −1 (1038)r + dim N(V DV ) = N (1039)For D ∈ EDM N −V T NDV N ≻ 0 ⇔ r = N −1 (1040)but −V DV ⊁ 0. The general fact 5.35 (confer (923))r ≤ min{n, N −1} (1041)is evident from (1029) but can be visualized in the example illustrated inFigure 115. 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.5.7.2 PrécisWe collect expressions for affine dimension r : 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 (960)= 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 (908) or G1=0 (912)= rankVN TDV N = rankV DV = rankV † N DV N = rankV N (VN TDV N)VNT= rank Λ (1128)([ ]) 0 1T= N −1 − dim N1 −D[ 0 1T= rank1 −D]− 2 (1049)(1042)⎫⎪⎬D ∈ EDM N⎪⎭5.35 rankX ≤ min{n , N}
5.7. EMBEDDING IN AFFINE HULL 4495.7.3 Eigenvalues of −V DV versus −V † N DV NSuppose for D ∈ EDM N we are given eigenvectors v i ∈ R N of −V DV andcorresponding eigenvalues λ∈ R N so that−V DV v i = λ i v i , i = 1... N (1043)From these we can determine the eigenvectors and eigenvalues of −V † N DV N :Defineν i V † N v i , λ i ≠ 0 (1044)Then we have:−V DV N V † N v i = λ i v i (1045)−V † N V DV N ν i = λ i V † N v i (1046)−V † N DV N ν i = λ i ν i (1047)the eigenvectors of −V † N DV N are given by (1044) while its correspondingnonzero eigenvalues are identical to those of −V DV although −V † N DV Nis not necessarily positive semidefinite. In contrast, −VN TDV N is positivesemidefinite but its nonzero eigenvalues are generally different.5.7.3.0.1 Theorem. EDM rank versus affine dimension r .[163,3] [184,3] [162,3] For D ∈ EDM N (confer (1201))1. r = rank(D) − 1 ⇔ 1 T D † 1 ≠ 0Points constituting a list X generating the polyhedron corresponding toD lie on the relative boundary of an r-dimensional circumhyperspherehavingdiameter = √ 2 ( 1 T D † 1 ) −1/2circumcenter = XD† 11 T D † 1(1048)2. r = rank(D) − 2 ⇔ 1 T D † 1 = 0There can be no circumhypersphere whose relative boundary containsa generating list for the corresponding polyhedron.3. In Cayley-Menger form [113,6.2] [88,3.3] [50,40] (5.11.2),([ ]) [ ]0 1T 0 1Tr = N −1 − dim N= rank − 2 (1049)1 −D 1 −DCircumhyperspheres exist for r< rank(D)−2. [347,7]⋄
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448 CHAPTER 5. EUCLIDEAN DISTANCE MATRIXBy conservation of dimension, (A.7.3.0.1)r + dim N(VNDV T N ) = N −1 (1038)r + dim N(V DV ) = N (1039)For D ∈ EDM N −V T NDV N ≻ 0 ⇔ r = N −1 (1040)but −V DV ⊁ 0. The general fact 5.35 (confer (923))r ≤ min{n, N −1} (1041)is evident from (1029) but can be visualized in the example illustrated inFigure 115. 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.5.7.2 PrécisWe collect expressions for affine dimension r : 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 (960)= 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 (908) or G1=0 (912)= rankVN TDV N = rankV DV = rankV † N DV N = rankV N (VN TDV N)VNT= rank Λ (1128)([ ]) 0 1T= N −1 − dim N1 −D[ 0 1T= rank1 −D]− 2 (1049)(1042)⎫⎪⎬D ∈ EDM N⎪⎭5.35 rankX ≤ min{n , N}