v2006.03.09 - Convex Optimization

Appendix EProjectionFor all A ∈ R m×n , the pseudoinverse [125,7.3, prob.9] [154,6.12, prob.19][88,5.5.4] [212, App.A]A † = limt→0 +(AT A + t I) −1 A T = limt→0 +AT (AA T + t I) −1 ∈ R n×m (1440)is a unique matrix having [166] [209,III.1, exer.1] E.1R(A † ) = R(A T ), R(A †T ) = R(A) (1441)N(A † ) = N(A T ), N(A †T ) = N(A) (1442)and satisfies the Penrose conditions: [244, Moore-Penrose] [127,1.3]1. AA † A = A2. A † AA † = A †3. (AA † ) T = AA †4. (A † A) T = A † AThe Penrose conditions are necessary and sufficient to establish thepseudoinverse whose principal action is to injectively map R(A) onto R(A T ).E.1 Proof of (1441) and (1442) is by singular value decomposition (A.6).2001 Jon Dattorro. CO&EDG version 03.09.2006. All rights reserved.Citation: Jon Dattorro, Convex Optimization & Euclidean Distance Geometry,Meboo Publishing USA, 2005.523