v2006.03.09 - Convex Optimization

5.5. EDM DEFINITION IN 11 T 3235.5.3 Faces of EDM cone5.5.3.1 Isomorphic facesIn high cardinality N , any set of EDMs made via (747) or (748) withparticular affine dimension r is isomorphic with any set admitting the sameaffine dimension but made in lower cardinality. We do not prove that here.5.5.3.2 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 (751)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 (149) 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 (737)where matrix V X (735) has orthogonal columns. For the same reason (1071)that zz T is an extreme direction of the positive semidefinite cone (2.9.2.3)for any particular nonzero vector z , there is no conic combination of distinctEDMs (each conically independent of Γ) equal to Γ .5.5.3.2.1 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