v2009.01.01 - Convex Optimization

396 CHAPTER 5. EUCLIDEAN DISTANCE MATRIX
By conservation of dimension, (A.7.3.0.1)
r + dim N(VNDV T N ) = N −1 (943)
r + dim N(V DV ) = N (944)
For D ∈ EDM N −V T NDV N ≻ 0 ⇔ r = N −1 (945)
but −V DV ⊁ 0. The general fact 5.32 (confer (831))
r ≤ min{n, N −1} (946)
is evident from (934) but can be visualized in the example illustrated in
Figure 92. There we imagine a vector from the origin to each point in the
list. Those three vectors are linearly independent in R 3 , but affine dimension
r is 2 because the three points lie in a plane. When that plane is translated
to the origin, it becomes the only subspace of dimension r=2 that can
contain the translated triangular polyhedron.
5.7.2 Précis
We collect expressions for affine dimension: for list X ∈ R n×N and Gram
matrix G∈ S N +
r
∆
= dim(P − α) = dim P = dim conv X
= dim(A − α) = dim A = dim aff X
= rank(X − x 1 1 T ) = rank(X − α c 1 T )
= rank Θ (865)
= rankXV N = rankXV = rankXV †T
N
= rankX , Xe 1 = 0 or X1=0
= rankVN TGV N = rankV GV = rankV † N GV N
= rankG , Ge 1 = 0 (815) or G1=0 (820)
= rankVN TDV N = rankV DV = rankV † N DV N = rankV N (VN TDV N)VN
T
= rank Λ (1033)
([ ]) 0 1
T
= N −1 − dim N
1 −D
[ 0 1
T
= rank
1 −D
]
− 2 (954)
(947)
⎫
⎪⎬
D ∈ EDM N
⎪⎭
5.32 rankX ≤ min{n , N}