v2010.10.26 - Convex Optimization

B.2. DOUBLET 639R([u v ])R(Π)= R([u v])0 0 N(Π)=v ⊥ ∩ u ⊥([ ]) vR N T= R([u v ]) ⊞ Nu Tv ⊥ ∩ u ⊥([ ]) vTNu T ⊞ R([u v ]) = R NFigure 157: The four fundamental subspaces [333,3.6] of a doubletΠ = uv T + vu T ∈ S N . Π(x)=(uv T + vu T )x is a linear bijective mappingfrom R([u v ]) to R([u v ]).Proof. (⇒) N(SW T )⊇ N(W T ) and R(SW T )⊆ R(S) are obvious.(⇐) Assume the existence of a left inverse B ∈ R k×N and a right inverseZ ∈ R N×k . B.5N(SW T ) = {x | SW T x = 0} ⊆ {x | BSW T x = 0} = N(W T ) (1629)R(SW T ) = {SW T x | x∈ R N } ⊇ {SW T Zy | Zy ∈ R N } = R(S) (1630)B.2 DoubletConsider a sum of two linearly independent square dyads, one a transpositionof the other:Π = uv T + vu T = [u v ] [ ]v Tu T = SW T ∈ S N (1631)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 (1632)By assumption of independence, a nonzero doublet has two nonzeroeigenvaluesλ 1 u T v + ‖uv T ‖ , λ 2 u T v − ‖uv T ‖ (1633)B.5 By counterexample, the theorem’s converse cannot be true; e.g., S = W = [1 0].