v2010.10.26 - Convex Optimization

636 APPENDIX B. SIMPLE MATRICESProof. Figure 156 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 ;u ∈ R(uv T )u ∈ N(uv T ) ⇔ v T u = 0R(uv T ) ∩ N(uv T ) = ∅(1615)B.1.0.1rank-one modificationFor A∈ R M×N , x∈ R N , y ∈ R M , and y T Ax≠0 [206,2.1]( )rank A − AxyT A= rank(A) − 1 (1616)y T AxIf A∈ R N×N is any nonsingular matrix and 1 + v T A −1 u ≠ 0, then[221, App.6] [393,2.3 prob.16] [153,4.11.2] (Sherman-Morrison)(A + uv T ) −1 = A −1 − A−1 uv T A −11 + v T A −1 u(1617)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 (1618)and the remaining eigenvalue is almost always positive;λ = u T u = tr(uu T ) > 0 unless u=0 (1619)The matrix [ Ψ uu T 1](1620)for example, is rank-1 positive semidefinite if and only if Ψ = uu T .