v2007.09.13 - Convex Optimization

B.4. AUXILIARY V -MATRICES 52918. V T = V = V N V † N = I − 1 N 11T ∈ S N(19. −V † N (11T − I)V N = I , 11 T − I ∈ EDM N)20. D = [d ij ] ∈ S N h (730)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 (1433)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 N22. D = [d ij ] ∈ S N h∑i,ji≠jd ij − N−1N∑d ii = 1 T D1 1 − trD Nitr(−V T N DV N) = ∑ jd 1j23. For Y ∈ S NB.4.3V (Y − δ(Y 1))V = Y − δ(Y 1)The skinny matrix⎡V ∆ W =⎢⎣Orthonormal auxiliary matrix V W−1 √N1 + −1N+ √ N−1N+ √ N.−1N+ √ N√−1−1N· · · √N−1N+ √ N......−1N+ √ N· · ·−1N+ √ N...−1N+ √ N...· · · 1 + −1N+ √ N.⎤∈ R N×N−1 (1434)⎥⎦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. [223]