v2007.09.13 - Convex Optimization

6.5. EDM DEFINITION IN 11 T 401and y=0 if and only if 1 is an eigenvector of EDM D . Scalar λ becomesan eigenvalue when corresponding eigenvector 1 exists. 6.5Then the particular dyad sum from (996)y1 T + 1y T + λ N 11T ∈ S N⊥c (998)must belong to the orthogonal complement of the geometric center subspace(p.610), whereas V X V T X ∈ SN c ∩ S N + (992) belongs to the positive semidefinitecone in the geometric center subspace.Proof. We validate eigenvector 1 and eigenvalue λ .(⇒) Suppose 1 is an eigenvector of EDM D . Then becauseit followsV T X 1 = 0 (999)D1 = 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 (1000)For some κ∈ R +δ(V X V T X ) T 1 = N κ = tr(V T X V X ) = ‖V X ‖ 2 F ⇒ δ(V X V T X ) = 1 N ‖V X ‖ 2 F1 (1001)so 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 (1002)1 is an eigenvector with corresponding eigenvalue λ . 6.5.1 Range of EDM DFromB.1.1 pertaining to linear independence of dyad sums: If the transposehalves of all the dyads in the sum (990) 6.6 make a linearly independent set,6.5 e.g., when X = I in EDM definition (705).6.6 Identifying columns V X∆= [v1 · · · v r ] , then V X V T X = ∑ iv i v T iis also a sum of dyads.