v2010.10.26 - Convex Optimization

700 APPENDIX E. PROJECTIONR np = AA † bR m{x}xR(A T )0N(A)x = A † px = A † b0 = A † (b − p)Ax = pR(A)p0bb −pN(A T ){b}Figure 161:(confer Figure 16) Pseudoinverse A † ∈ R n×m action: [331, p.449]Component of b in N(A T ) maps to 0, while component of b in R(A) mapsto rowspace R(A T ). For any A∈ R m×n , inversion is bijective ∀p ∈ R(A).x=A † b ⇔ x∈R(A T ) & b−Ax⊥R(A) ⇔ x⊥ N(A) & b−Ax∈ N(A T ). [48]The following relations reliably hold without qualification:a. A T† = A †Tb. A †† = Ac. (AA T ) † = A †T A †d. (A T A) † = A † A †Te. (AA † ) † = AA †f. (A † A) † = A † AYet for arbitrary A,B it is generally true that (AB) † ≠ B † A † :E.0.0.0.1 Theorem. Pseudoinverse of product. [169] [57] [241, exer.7.23]For A∈ R m×n and B ∈ R n×k (AB) † = B † A † (1873)if and only ifR(A T AB) ⊆ R(B) and R(BB T A T ) ⊆ R(A T ) (1874)⋄