v2007.09.13 - Convex Optimization

580 APPENDIX E. PROJECTIONThe 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. [118] [42] [175, exer.7.23]For A∈ R m×n and B ∈ R n×k (AB) † = B † A † (1637)if and only ifR(A T AB) ⊆ R(B) and R(BB T A T ) ⊆ R(A T ) (1638)⋄U T = U † for orthonormal (including the orthogonal) matrices U . So, fororthonormal matrices U,Q and arbitrary A(UAQ T ) † = QA † U T (1639)E.0.0.0.2Prove:Exercise. Kronecker inverse.(A ⊗ B) † = A † ⊗ B † (1640)E.0.1Logical deductionsWhen A is invertible, A † = A −1 ; so A † A = AA † = I . Otherwise, forA∈ R m×n [102,5.3.3.1] [175,7] [218]g. A † A = I , A † = (A T A) −1 A T , rankA = nh. AA † = I , A † = A T (AA T ) −1 , rankA = mi. A † Aω = ω , ω ∈ R(A T )j. AA † υ = υ , υ ∈ R(A)k. A † A = AA † , A normall. A k† = A †k , A normal, k an integer