v2009.01.01 - Convex Optimization

580 APPENDIX B. SIMPLE MATRICES
Proof. Figure 132 shows the four fundamental subspaces for the dyad.
Linear operator Ψ : R N →R M provides a map between vector spaces that
remain distinct when M =N ;
B.1.0.1
rank-one modification
u ∈ R(uv T )
u ∈ N(uv T ) ⇔ v T u = 0
R(uv T ) ∩ N(uv T ) = ∅
(1501)
If A∈ R N×N is any nonsingular matrix and 1 + v T A −1 u ≠ 0, then
[192, App.6] [344,2.3, prob.16] [127,4.11.2] (Sherman-Morrison)
B.1.0.2
dyad symmetry
(A + uv T ) −1 = A −1 − A−1 uv T A −1
1 + v T A −1 u
(1502)
In the specific circumstance that v = u , then uu T ∈ R N×N is symmetric,
rank-one, and positive semidefinite having exactly N −1 0-eigenvalues. In
fact, (Theorem A.3.1.0.7)
uv T ≽ 0 ⇔ v = u (1503)
and the remaining eigenvalue is almost always positive;
λ = u T u = tr(uu T ) > 0 unless u=0 (1504)
The matrix [ Ψ u
u T 1
for example, is rank-1 positive semidefinite if and only if Ψ = uu T .
]
(1505)
B.1.1 Dyad independence
Now we consider a sum of dyads like (1492) as encountered in diagonalization
and singular value decomposition:
( k∑
)
k∑
R s i wi
T = R ( )
k∑
s i wi
T = R(s i ) ⇐ w i ∀i are l.i. (1506)
i=1
i=1
i=1