v2009.01.01 - Convex Optimization

352 CHAPTER 5. EUCLIDEAN DISTANCE MATRIX
Now we develop some invaluable concepts, moving toward a link of the
Euclidean metric properties to matrix criteria.
5.4 EDM definition
Ascribe points in a list {x l ∈ R n , l=1... N} to the columns of a matrix
X = [x 1 · · · x N ] ∈ R n×N (68)
where N is regarded as cardinality of list X . When matrix D =[d ij ] is an
EDM, its entries must be related to those points constituting the list by the
Euclidean distance-square: for i,j=1... N (A.1.1 no.27)
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 xi
j
−I I x j
= vec(X) T (Φ ij ⊗ I) vecX = 〈Φ ij , X T X〉
(794)
where
⎡
vec X = ⎢
⎣
⎤
x 1
x 2
⎥
. ⎦ ∈ RnN (795)
x N
and where ⊗ signifies Kronecker product (D.1.2.1). Φ ij ⊗ I is positive
semidefinite (1384) having 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 (796)
i
= (e i − e j )(e i − e j ) T
where {e i ∈ R N , i=1... N} is the set of standard basis vectors. Thus each
entry d ij is a convex quadratic function (A.4.0.0.1) of vec X (30). [266,6]