v2006.03.09 - Convex Optimization

464 APPENDIX B. SIMPLE MATRICESB.1.0.1rank-one modificationIf A∈ R N×N is any nonsingular matrix and 1+v T A −1 u≠0, then [136, App.6][261,2.3, prob.16] [82,4.11.2] (Sherman-Morrison)(A + uv T ) −1 = A −1 − A−1 uv T A −11 + v T A −1 u(1203)B.1.0.2dyad symmetryIn 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 (1204)and the remaining eigenvalue is almost always positive;λ = u T u = tr(uu T ) > 0 unless u=0 (1205)The matrix [ Ψ uu T 1for example, is rank-1 positive semidefinite if and only if Ψ = uu T .](1206)B.1.1Dyad independenceNow we consider a sum of dyads like (1193) 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. (1207)i=1i=1i=1range of the summation is the vector sum of ranges. B.3 (Theorem B.1.1.1.1)Under the assumption the dyads are linearly independent (l.i.), then thevector sums are unique (p.610): for {w i } l.i. and {s i } l.i.( k∑)R s i wiT = R ( ) ( )s 1 w1 T ⊕ ... ⊕ R sk wk T = R(s1 ) ⊕ ... ⊕ R(s k ) (1208)i=1B.3 Move of range R to inside the summation depends on linear independence of {w i }.