v2009.01.01 - Convex Optimization

Appendix E
Projection
For any A∈ R m×n , the pseudoinverse [176,7.3, prob.9] [215,6.12, prob.19]
[134,5.5.4] [287, App.A]
A †
∆ = lim
t→0 +(AT A + t I) −1 A T = lim
t→0 +AT (AA T + t I) −1 ∈ R n×m (1745)
is a unique matrix solving minimize ‖AX − I‖ 2 F . For any t > 0
X
I − A(A T A + t I) −1 A T = t(AA T + t I) −1 (1746)
Equivalently, pseudoinverse A † is that unique matrix satisfying the
Moore-Penrose conditions: [178,1.3] [326]
1. AA † A = A 3. (AA † ) T = AA †
2. A † AA † = A † 4. (A † A) T = A † A
which are necessary and sufficient to establish the pseudoinverse whose
principal action is to injectively map R(A) onto R(A T ) (Figure 137). Taken
rowwise, these conditions are respectively necessary and sufficient for AA †
to be the orthogonal projector on R(A) , and for A † A to be the orthogonal
projector on R(A T ).
Range and nullspace of the pseudoinverse [230] [285,III.1, exer.1]
R(A † ) = R(A T ), R(A †T ) = R(A) (1747)
N(A † ) = N(A T ), N(A †T ) = N(A) (1748)
can be derived by singular value decomposition (A.6).
2001 Jon Dattorro. CO&EDG version 2009.01.01. All rights reserved.
Citation: Jon Dattorro, Convex Optimization & Euclidean Distance Geometry,
Meboo Publishing USA, 2005.
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