v2010.10.26 - Convex Optimization

510 CHAPTER 6. CONE OF DISTANCE MATRICES6.4.3.2 extreme directions 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 (1209)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 (186) 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 (1185)where matrix V X (1186) has orthogonal columns. For the same reason (1467)that zz T is an extreme direction of the positive semidefinite cone (2.9.2.7)for any particular nonzero vector z , there is no conic combination of distinctEDMs (each conically independent of Γ (2.10)) equal to Γ . 6.4.3.2.1 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 holdsD = δ ( −V DV 2) 1 1 T + 1δ ( −V DV 2) 1 T ( )− 2 −V DV12(999)By diagonalization −V DV 1 2 QΛQT ∈ S N c(A.5.1) we may write( N) (∑N)∑ T∑D = δ λ i q i qiT 1 T + 1δ λ i q i qiT − 2 N λ i q i qiTi=1i=1i=1∑= N ( )λ i δ(qi qi T )1 T + 1δ(q i qi T ) T − 2q i qiTi=1(1210)where q i is the i th eigenvector of −V DV 1 arranged columnar in orthogonal2matrixQ = [q 1 q 2 · · · q N ] ∈ R N×N (400)