v2007.09.17 - Convex Optimization

522 APPENDIX B. SIMPLE MATRICESProof. Figure 113 shows the four fundamental subspaces for the dyad.Linear operator Ψ : R N →R M provides a map between vector spaces thatremain distinct when M =N ;B.1.0.1rank-one modificationu ∈ R(uv T )u ∈ N(uv T ) ⇔ v T u = 0R(uv T ) ∩ N(uv T ) = ∅(1400)If A∈ R N×N is any nonsingular matrix and 1+v T A −1 u≠0, then [162, App.6][301,2.3, prob.16] [103,4.11.2] (Sherman-Morrison)B.1.0.2dyad symmetry(A + uv T ) −1 = A −1 − A−1 uv T A −11 + v T A −1 u(1401)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. Infact, (Theorem A.3.1.0.7)uv T ≽ 0 ⇔ v = u (1402)and the remaining eigenvalue is almost always positive;λ = u T u = tr(uu T ) > 0 unless u=0 (1403)The matrix [ Ψ uu T 1for example, is rank-1 positive semidefinite if and only if Ψ = uu T .](1404)B.1.1 Dyad independenceNow we consider a sum of dyads like (1391) as encountered in diagonalizationand singular value decomposition:( k∑)k∑R s i wiT = R ( )k∑s i wiT = R(s i ) ⇐ w i ∀i are l.i. (1405)i=1i=1i=1