v2009.01.01 - Convex Optimization

B.4. AUXILIARY V -MATRICES 589
18. 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 (823)
tr(−V DV ) = tr(−V D) = tr(−V † N DV N) = 1 N 1T D 1 = 1 N tr(11T D) = 1 N
Any elementary matrix E ∈ S N of the particular form
∑
d ij
i,j
E = k 1 I − k 2 11 T (1539)
where k 1 , k 2 ∈ R , B.7 will make tr(−ED) proportional to ∑ d ij .
21. D = [d ij ] ∈ S N
tr(−V DV ) = 1 N
22. D = [d ij ] ∈ S N h
∑
i,j
i≠j
d ij − N−1
N
∑
d ii = 1 T D1 1 − trD N
i
tr(−V T N DV N) = ∑ j
d 1j
23. For Y ∈ S N
B.4.3
V (Y − δ(Y 1))V = Y − δ(Y 1)
The skinny matrix
⎡
V ∆ W =
⎢
⎣
Orthonormal auxiliary matrix V W
−1 √
N
1 + −1
N+ √ N
−1
N+ √ N
.
−1
N+ √ N
√−1
−1
N
· · · √
N
−1
N+ √ N
...
...
−1
N+ √ N
· · ·
−1
N+ √ N
...
−1
N+ √ N
...
· · · 1 + −1
N+ √ N
.
⎤
∈ R N×N−1 (1540)
⎥
⎦
B.7 If k 1 is 1−ρ while k 2 equals −ρ∈R , then all eigenvalues of E for −1/(N −1) < ρ < 1
are guaranteed positive and therefore E is guaranteed positive definite. [261]