v2007.11.26 - Convex Optimization

B.2. DOUBLET 539R([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 119: Four fundamental subspaces [253,3.6] 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 ]).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 (1441)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 (1442)By assumption of independence, a nonzero doublet has two nonzeroeigenvaluesλ 1 ∆ = u T v + ‖uv T ‖ , λ 2 ∆ = u T v − ‖uv T ‖ (1443)where λ 1 > 0 >λ 2 , with corresponding eigenvectorsx 1 ∆ = u‖u‖ + v‖v‖ , x 2∆= u‖u‖ −v‖v‖(1444)spanning the doublet range. Eigenvalue λ 1 cannot be 0 unless u and v haveopposing directions, but that is antithetical since then the dyads would nolonger be independent. Eigenvalue λ 2 is 0 if and only if u and v share thesame direction, again antithetical. Generally we have λ 1 > 0 and λ 2 < 0, soΠ is indefinite.