v2009.01.01 - Convex Optimization

6.5. EDM DEFINITION IN 11 T 465
6.5.3.2 Extreme directions of EDM cone
In particular, extreme directions (2.8.1) of EDM N correspond to affine
dimension r = 1 and are simply represented: for any particular cardinality
N ≥ 2 (2.8.2) and each and every nonzero vector z in N(1 T )
Γ ∆ = (z ◦ z)1 T + 1(z ◦ z) T − 2zz T ∈ EDM N
= δ(zz T )1 T + 1δ(zz T ) T − 2zz T (1117)
is an extreme direction corresponding to a one-dimensional face of the EDM
cone EDM N that is a ray in isomorphic subspace R N(N−1)/2 .
Proving this would exercise the fundamental definition (168) of extreme
direction. Here is a sketch: Any EDM may be represented
D(V X ) ∆ = δ(V X V T X )1 T + 1δ(V X V T X ) T − 2V X V T X ∈ EDM N (1093)
where matrix V X (1094) has orthogonal columns. For the same reason (1371)
that zz T is an extreme direction of the positive semidefinite cone (2.9.2.4)
for any particular nonzero vector z , there is no conic combination of distinct
EDMs (each conically independent of Γ) equal to Γ .
6.5.3.2.1 Example. Biorthogonal expansion of an EDM.
(confer2.13.7.1.1) When matrix D belongs to the EDM cone, nonnegative
coordinates for biorthogonal expansion are the eigenvalues λ∈ R N of
−V DV 1 : For any D ∈ 2 SN h it holds
D = δ ( −V DV 2) 1 1 T + 1δ ( −V DV 2) 1 T ( )
− 2 −V DV
1
2
(904)
By diagonalization −V DV 1 2
∆
= QΛQ T ∈ S N c (A.5.2) we may write
( N
) (
∑
N
)
∑ T
∑
D = δ λ i q i qi
T 1 T + 1δ λ i q i qi
T − 2 N λ i q i qi
T
i=1
i=1
i=1
∑
= N ( )
λ i δ(qi qi T )1 T + 1δ(q i qi T ) T − 2q i qi
T
i=1
(1118)
where q i is the i th eigenvector of −V DV 1 arranged columnar in orthogonal
2
matrix
Q = [q 1 q 2 · · · q N ] ∈ R N×N (354)