v2007.09.17 - Convex Optimization

298 CHAPTER 5. EUCLIDEAN DISTANCE MATRIXNow we develop some invaluable concepts, moving toward a link of theEuclidean metric properties to matrix criteria.5.4 EDM definitionAscribe points in a list {x l ∈ R n , l=1... N} to the columns of a matrixX = [x 1 · · · x N ] ∈ R n×N (65)where N is regarded as cardinality of list X . When matrix D =[d ij ] is anEDM, its entries must be related to those points constituting the list by theEuclidean distance-square: for i,j=1... N (A.1.1 no.23)d ij = ‖x i − x j ‖ 2 = (x i − x j ) T (x i − x j ) = ‖x i ‖ 2 + ‖x j ‖ 2 − 2x T ix j= [ ] [ ] [ ]x T i x T I −I xij−I I x j= vec(X) T (Φ ij ⊗ I) vec X = 〈Φ ij , X T X〉(705)where⎡vec X = ⎢⎣⎤x 1x 2⎥. ⎦ ∈ RnN (706)x Nand where Φ ij ⊗ I has I ∈ S n in its ii th and jj th block of entries while−I ∈ S n fills its ij th and ji th block; id est,Φ ij ∆ = δ((e i e T j + e j e T i )1) − (e i e T j + e j e T i ) ∈ S N += e i e T i + e j e T j − e i e T j − e j e T (707)i= (e i − e j )(e i − e j ) Twhere {e i ∈ R N , i=1... N} is the set of standard basis vectors, and ⊗signifies the Kronecker product (D.1.2.1). Thus each entry d ij is a convexquadratic function [46,3,4] of vec X (30). [230,6]