v2009.01.01 - Convex Optimization

640 APPENDIX E. PROJECTION
R n
p = AA † b
R m
{x}
x
R(A T )
0
N(A)
x = A † p
x = A † b
0 = A † (b − p)
Ax = p
R(A)
p
0
b
b − p
N(A T )
{b}
Figure 137: Action of pseudoinverse A † ∈ R n×m : [287, p.449] Component of
vector b in N(A T ) maps to origin, while component of b in R(A) maps to
rowspace R(A T ). For any A∈ R m×n , inversion is bijective ∀p ∈ R(A).
The following relations reliably hold without qualification:
a. A T† = A †T
b. A †† = A
c. (AA T ) † = A †T A †
d. (A T A) † = A † A †T
e. (AA † ) † = AA †
f. (A † A) † = A † A
Yet for arbitrary A,B it is generally true that (AB) † ≠ B † A † :
E.0.0.0.1 Theorem. Pseudoinverse of product. [143] [49] [208, exer.7.23]
For A∈ R m×n and B ∈ R n×k (AB) † = B † A † (1749)
if and only if
R(A T AB) ⊆ R(B) and R(BB T A T ) ⊆ R(A T ) (1750)
⋄