v2009.01.01 - Convex Optimization

B.2. DOUBLET 583
R([u v ])
R(Π)= R([u v])
0 0
N(Π)=v ⊥ ∩ u ⊥
([ ]) v
R N T
= R([u v ]) ⊕ N
u T
v ⊥ ∩ u ⊥
([ ]) v
T
N
u T ⊕ R([u v ]) = R N
Figure 133: Four fundamental subspaces [289,3.6] of a doublet
Π = uv T + vu T ∈ S N . Π(x) = (uv T + vu T )x is a linear bijective mapping
from R([u v ]) to R([u v ]).
B.2 Doublet
Consider a sum of two linearly independent square dyads, one a transposition
of the other:
Π = uv T + vu T = [ u v ] [ ]
v T
u T = SW T ∈ S N (1516)
where u,v ∈ R N . Like the dyad, a doublet can be 0 only when u or v is 0;
Π = uv T + vu T = 0 ⇔ u = 0 or v = 0 (1517)
By assumption of independence, a nonzero doublet has two nonzero
eigenvalues
λ 1 ∆ = u T v + ‖uv T ‖ , λ 2 ∆ = u T v − ‖uv T ‖ (1518)
where λ 1 > 0 >λ 2 , with corresponding eigenvectors
x 1 ∆ = u
‖u‖ + v
‖v‖ , x 2
∆
= u
‖u‖ −
v
‖v‖
(1519)
spanning the doublet range. Eigenvalue λ 1 cannot be 0 unless u and v have
opposing directions, but that is antithetical since then the dyads would no
longer be independent. Eigenvalue λ 2 is 0 if and only if u and v share the
same direction, again antithetical. Generally we have λ 1 > 0 and λ 2 < 0, so
Π is indefinite.