v2007.09.13 - Convex Optimization

300 CHAPTER 5. EUCLIDEAN DISTANCE MATRIX5.4.2.1 First point at originAssume the first point x 1 in an unknown list X resides at the origin;Xe 1 = 0 ⇔ Ge 1 = 0 (719)Consider the symmetric translation (I − 1e T 1 )D(G)(I − e 1 1 T ) that shiftsthe first row and column of D(G) to the origin; setting Gram-form EDMoperator D(G) = D for convenience,− ( D − (De 1 1 T + 1e T 1D) + 1e T 1De 1 1 T) 12 = G − (Ge 11 T + 1e T 1G) + 1e T 1Ge 1 1 Twheree 1 ∆ =⎡⎢⎣10.0⎤(720)⎥⎦ (721)is the first vector from the standard basis. Then it follows for D ∈ S N hG = − ( D − (De 1 1 T + 1e T 1D) ) 12 , x 1 = 0= − [ 0 √ ] TD [ √ ]2V N 0 1 2VN2[ ] 0 0T=0 −VN TDV NVN TGV N = −VN TDV N 1 2∀X(722)whereI − e 1 1 T =[0 √ 2V N](723)is a projector nonorthogonally projecting (E.1) onS N 1 = {G∈ S N | Ge 1 = 0}{ [0 √ ] T [ √ ]= 2VN Y 0 2VN | Y ∈ SN} (1765)in the Euclidean sense. From (722) we get sufficiency of the first matrixcriterion for an EDM proved by Schoenberg in 1935; [232] 5.75.7 From (712) we know R(V N )= N(1 T ) , so (724) is the same as (700). In fact, anymatrix V in place of V N will satisfy (724) whenever R(V )= R(V N )= N(1 T ). But V N isthe matrix implicit in Schoenberg’s seminal exposition.