v2010.10.26 - Convex Optimization

B.4. AUXILIARY V -MATRICES 64514. [V N1 √21 ] −1 =[ ]V†N√2N 1T15. V † N = √ 2 [ − 1 N 1 I − 1 N 11T] ∈ R N−1×N ,16. V † N 1 = 017. V † N V N = I18. V T = V = V N V † N = I − 1 N 11T ∈ S N(I −1N 11T ∈ S N−1)19. −V † N (11T − I)V N = I ,(11 T − I ∈ EDM N)20. D = [d ij ] ∈ S N h (915)tr(−V DV ) = tr(−V D) = tr(−V † N DV N) = 1 N 1T D 1 = 1 N tr(11T D) = 1 NAny elementary matrix E ∈ S N of the particular form∑d iji,jE = k 1 I − k 2 11 T (1654)where k 1 , k 2 ∈ R , B.7 will make tr(−ED) proportional to ∑ d ij .21. D = [d ij ] ∈ S Ntr(−V DV ) = 1 N∑i,ji≠jd ij − N−1N∑d ii = 1 N 1T D 1 − trDi22. D = [d ij ] ∈ S N htr(−V T N DV N) = ∑ jd 1j23. For Y ∈ S NV (Y − δ(Y 1))V = Y − δ(Y 1)B.7 If k 1 is 1−ρ while k 2 equals −ρ∈R , then all eigenvalues of E for −1/(N −1) < ρ < 1are guaranteed positive and therefore E is guaranteed positive definite. [301]