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v2006.03.09 - Convex Optimization

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524 APPENDIX E. PROJECTIONThe following relations are reliably true 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. [97] [149, exer.7.23]For A∈ R m×n and B ∈ R n×k (AB) † = B † A † (1443)if and only ifR(A T AB) ⊆ R(B) and R(BB T A T ) ⊆ R(A T ) (1444)For orthogonal matrices U,Q and arbitrary A [209,III.1]E.0.1Logical deductions(UAQ T ) † = QA † U T (1445)When A is invertible, A † = A −1 , of course; so A † A = AA † = I . Otherwise,for A∈ R m×n [82,5.3.3.1] [149,7] [186]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 integerWhen A is symmetric, A † is symmetric and (A.6)A ≽ 0 ⇔ A † ≽ 0 (1446)⋄

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