v2004.06.19 - Convex Optimization

5.2. EDM DEFINITION IN 11 T 207X = I in EDM definition (333).Formula (594) can be validated by substituting (595); 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 (596)is simply the standard EDM definition (333) where X T X has been replacedwith the subcompact singular value decomposition (§A.6.2) 5.2V X V T X ≡ V T X T XV (597)Then the inner product VX TV X is an r ×r diagonal matrix Σ of nonzerosingular values. Next we validate eigenvector 1 and eigenvalue λ .(=⇒) Suppose 1 is an eigenvector of EDM D . Then becauseV T X 1 = 0 (598)it followsD1 = δ(V X V T X )1T 1 + 1δ(V X V T X )T 1 = N δ(V X V T X ) + ‖V X ‖ 2 F 1⇒ δ(V X V T X ) ∝ 1 (599)δ(V X VX T)T 1 = Nκ = tr(VX TV X) = ‖V X ‖ 2 F ⇒ δ(V X V X T )=κ1 and y=0.(⇐=) Now suppose δ(V X VX T)= λ 1 ; id est, y=0. Then2ND = λ N 11T − 2V X V T X ∈ EDM N (600)1 is an eigenvector with corresponding eigenvalue λ . ∆5.2 Subcompact SVD: V X VXT =QΣ 1/2 Σ 1/2 Q T ≡V T X T XV . So VXT is not necessarily XV(§4.5.1.0.1), although affine dimension r = rank(VX T ) = rank(XV ). (436)