v2010.10.26 - Convex Optimization

Appendix EProjectionFor any A∈ R m×n , the pseudoinverse [202,7.3 prob.9] [250,6.12 prob.19][159,5.5.4] [331, App.A]A † limt→0 +(AT A + t I) −1 A T = limt→0 +AT (AA T + t I) −1 ∈ R n×m (1869)is a unique matrix from the optimal solution set to minimize ‖AX − I‖ 2 FX(3.6.0.0.2). For any t > 0I − A(A T A + t I) −1 A T = t(AA T + t I) −1 (1870)Equivalently, pseudoinverse A † is that unique matrix satisfying theMoore-Penrose conditions: [204,1.3] [373]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 ) (Figure 161).Conditions 1 and 3 are necessary and sufficient for AA † to be the orthogonalprojector on R(A) , while conditions 2 and 4 hold iff A † A is the orthogonalprojector on R(A T ).Range and nullspace of the pseudoinverse [267] [328,III.1 exer.1]R(A † ) = R(A T ), R(A †T ) = R(A) (1871)N(A † ) = N(A T ), N(A †T ) = N(A) (1872)can be derived by singular value decomposition (A.6).2001 Jon Dattorro. co&edg version 2010.10.26. All rights reserved.citation: Dattorro, Convex Optimization & Euclidean Distance Geometry,Mεβoo Publishing USA, 2005, v2010.10.26.699