v2006.03.09 - Convex Optimization

5.5. EDM DEFINITION IN 11 T 3175.5 EDM definition in 11 TAny EDM D corresponding to affine dimension r has representation(confer (460))D(V X , y) ∆ = y1 T + 1y T − 2V X V T X + λ N 11T ∈ EDM N (734)where R(V X ∈ R N×r )⊆ N(1 T ) ,V T X V X = δ 2 (V T X V X ) , V X is full-rank with orthogonal columns, (735)λ ∆ = 2‖V X ‖ 2 F , and y ∆ = δ(V X V T X ) − λ2N 1 (736)where y=0 if and only if 1 is an eigenvector of EDM D . [110,2] Scalar λbecomes an eigenvalue when corresponding eigenvector 1 exists; e.g., whenX = I in EDM definition (460).Formula (734) can be validated by substituting (736); we findD(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)is simply the standard EDM definition (460) where X T X has been replacedwith the subcompact singular value decomposition (A.6.2)V X V T X ≡ V T X T XV (738)Then inner product VX TV Xsingular values. 5.3is an r×r diagonal matrix Σ of nonzero∆5.3 Subcompact SVD: V X VXT = QΣ 1/2 Σ 1/2 Q T ≡ V T X T XV . So VX T is not necessarilyXV (4.5.1.0.1), although affine dimension r = rank(VX T ) = rank(XV ). (581)