v2009.01.01 - Convex Optimization

346 CHAPTER 5. EUCLIDEAN DISTANCE MATRIX
√
5
2
1
⎡
D = ⎣
1 2 3
0 1 5
1 0 4
5 4 0
⎤
⎦
1
2
3
Figure 92: Convex hull of three points (N = 3) is shaded in R 3 (n=3). Dotted
lines are imagined vectors to points.
5.1 EDM
Euclidean space R n is a finite-dimensional real vector space having an inner
product defined on it, hence a metric as well. [197,3.1] A Euclidean distance
matrix, an EDM in R N×N
+ , is an exhaustive table of distance-square d ij
between 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 〉 (786)
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 13
d 21 d 22 d 23
⎦ =
d 31 d 32 d 33
⎡
⎣
0 d 12 d 13
d 12 0 d 23
d 13 d 23 0
⎤
⎡
⎦ = ⎣
0 1 5
1 0 4
5 4 0
⎤
⎦ (787)
Matrix D has N 2 entries but only N(N −1)/2 pieces of information. In
Figure 92 are shown three points in R 3 that can be arranged in a list to