v2007.09.13 - Convex Optimization

Appendix EProjectionFor any A∈ R m×n , the pseudoinverse [149,7.3, prob.9] [181,6.12, prob.19][109,5.5.4] [247, App.A]A †∆ = limt→0 +(AT A + t I) −1 A T = limt→0 +AT (AA T + t I) −1 ∈ R n×m (1633)is a unique matrix solving minimize ‖AX − I‖ 2 F . For any t > 0XI − A(A T A + t I) −1 A T = t(AA T + t I) −1 (1634)Equivalently, pseudoinverse A † is that unique matrix satisfying theMoore-Penrose conditions: [151,1.3] [279]1. AA † A = A 3. (AA † ) T = AA †2. A † AA † = A † 4. (A † A) T = A † Awhich are necessary and sufficient to establish the pseudoinverse whoseprincipal action is to injectively map R(A) onto R(A T ). Taken rowwise,these conditions are respectively necessary and sufficient for AA † to be theorthogonal projector on R(A) , and for A † A to be the orthogonal projectoron R(A T ).Range and nullspace of the pseudoinverse [194] [244,III.1, exer.1]R(A † ) = R(A T ), R(A †T ) = R(A) (1635)N(A † ) = N(A T ), N(A †T ) = N(A) (1636)can be derived by singular value decomposition (A.6).2001 Jon Dattorro. CO&EDG version 2007.09.13. All rights reserved.Citation: Jon Dattorro, Convex Optimization & Euclidean Distance Geometry,Meboo Publishing USA, 2005.579