v2007.09.17 - Convex Optimization

5.7. EMBEDDING IN AFFINE HULL 3415.7.3 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 (852)From these we can determine the eigenvectors and eigenvalues of −V † N DV N :Defineν i ∆ = V † N v i , λ i ≠ 0 (853)Then we have:−V DV N V † N v i = λ i v i (854)−V † N V DV N ν i = λ i V † N v i (855)−V † N DV N ν i = λ i ν i (856)the eigenvectors of −V † N DV N are given by (853) 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 .[113,3] [133,3] [112,3] For D ∈ EDM N (confer (1010))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(857)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 [77,6.2] [60,3.3] [37,40] (5.11.2),([ ]) [ ]0 1T 0 1Tr = N −1 − dim N= rank − 2 (858)1 −D 1 −DCircumhyperspheres exist for r< rank(D)−2. [261,7]⋄