v2007.09.13 - Convex Optimization

406 CHAPTER 6. EDM CONE6.5.3 Faces of EDM cone6.5.3.0.1 Exercise. Isomorphic faces.Prove that in high cardinality N , any set of EDMs made via (1007) or (1008)with particular affine dimension r is isomorphic with any set admitting thesame affine dimension but made in lower cardinality.6.5.3.1 Extreme direction of EDM coneIn particular, extreme directions (2.8.1) of EDM N correspond to affinedimension r = 1 and are simply represented: for any particular cardinalityN ≥ 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 (1011)is an extreme direction corresponding to a one-dimensional face of the EDMcone EDM N that is a ray in isomorphic subspace R N(N−1)/2 .Proving this would exercise the fundamental definition (155) of extremedirection. Here is a sketch: Any EDM may be representedD(V X ) ∆ = δ(V X V T X )1 T + 1δ(V X V T X ) T − 2V X V T X ∈ EDM N (990)where matrix V X (991) has orthogonal columns. For the same reason (1268)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 distinctEDMs (each conically independent of Γ) equal to Γ .6.5.3.1.1 Example. Biorthogonal expansion of an EDM.(confer2.13.7.1.1) When matrix D belongs to the EDM cone, nonnegativecoordinates for biorthogonal expansion are the eigenvalues λ∈ R N of−V DV 1 : For any D ∈ 2 SN h it holds