v2010.10.26 - Convex Optimization

396 CHAPTER 5. EUCLIDEAN DISTANCE MATRIX√521⎡D = ⎣1 2 30 1 51 0 45 4 0⎤⎦123Figure 115: Convex hull of three points (N = 3) is shaded in R 3 (n=3).Dotted lines are imagined vectors to points whose affine dimension is 2.5.1 EDMEuclidean space R n is a finite-dimensional real vector space having an innerproduct defined on it, inducing a metric. [227,3.1] A Euclidean distancematrix, an EDM in R N×N+ , is an exhaustive table of distance-square d ijbetween points taken by pair from a list of N points {x l , l=1... N} in R n ;the squared metric, the measure of distance-square:d ij = ‖x i − x j ‖ 2 2 〈x i − x j , x i − x j 〉 (879)Each point is labelled ordinally, hence the row or column index of an EDM,i or j =1... N , individually addresses all the points in the list.Consider the following example of an EDM for the case N = 3 :D = [d ij ] =⎡⎣⎤d 11 d 12 d 13d 21 d 22 d 23⎦ =d 31 d 32 d 33⎡⎣0 d 12 d 13d 12 0 d 23d 13 d 23 0⎤⎡⎦ = ⎣0 1 51 0 45 4 0⎤⎦ (880)Matrix D has N 2 entries but only N(N −1)/2 pieces of information. InFigure 115 are shown three points in R 3 that can be arranged in a list