v2009.01.01 - Convex Optimization

5.7. EMBEDDING IN AFFINE HULL 397
5.7.3 Eigenvalues of −V DV versus −V † N DV N
Suppose for D ∈ EDM N we are given eigenvectors v i ∈ R N of −V DV and
corresponding eigenvalues λ∈ R N so that
−V DV v i = λ i v i , i = 1... N (948)
From these we can determine the eigenvectors and eigenvalues of −V † N DV N :
Define
ν i ∆ = V † N v i , λ i ≠ 0 (949)
Then we have:
−V DV N V † N v i = λ i v i (950)
−V † N V DV N ν i = λ i V † N v i (951)
−V † N DV N ν i = λ i ν i (952)
the eigenvectors of −V † N DV N are given by (949) while its corresponding
nonzero eigenvalues are identical to those of −V DV although −V † N DV N
is not necessarily positive semidefinite. In contrast, −VN TDV N is positive
semidefinite but its nonzero eigenvalues are generally different.
5.7.3.0.1 Theorem. EDM rank versus affine dimension r .
[137,3] [158,3] [136,3] For D ∈ EDM N (confer (1109))
1. r = rank(D) − 1 ⇔ 1 T D † 1 ≠ 0
Points constituting a list X generating the polyhedron corresponding to
D lie on the relative boundary of an r-dimensional circumhypersphere
having
diameter = √ 2 ( 1 T D † 1 ) −1/2
circumcenter = XD† 1
1 T D † 1
(953)
2. r = rank(D) − 2 ⇔ 1 T D † 1 = 0
There can be no circumhypersphere whose relative boundary contains
a generating list for the corresponding polyhedron.
3. In Cayley-Menger form [96,6.2] [77,3.3] [44,40] (5.11.2),
([ ]) [ ]
0 1
T 0 1
T
r = N −1 − dim N
= rank − 2 (954)
1 −D 1 −D
Circumhyperspheres exist for r< rank(D)−2. [301,7]
⋄