Greville's Method for Preconditioning Least Squares ... - Projects
Greville's Method for Preconditioning Least Squares ... - Projects
Greville's Method for Preconditioning Least Squares ... - Projects
You also want an ePaper? Increase the reach of your titles
YUMPU automatically turns print PDFs into web optimized ePapers that Google loves.
5<br />
we can express A † i<br />
in a unified <strong>for</strong>m <strong>for</strong> general matrices as<br />
and we have<br />
A † i = A† i−1 + 1 f i<br />
(e i − k i )v T i , (3.5)<br />
A † =<br />
n∑<br />
i=1<br />
If we define matrices K ∈ R n×n , V ∈ R m×n , F ∈ R n×n as<br />
we obtain a matrix factorization of A † as follows.<br />
1<br />
f i<br />
(e i − k i )v T i . (3.6)<br />
K = [k 1 , . . . , k n], (3.7)<br />
V = [v 1 , . . . , v n], (3.8)<br />
⎡<br />
⎤<br />
f 1 · · · 0<br />
⎢<br />
F =<br />
.<br />
⎣ 0 ..<br />
⎥<br />
0 ⎦ , (3.9)<br />
0 · · · f n<br />
Theorem 2 Let A ∈ R m×n and rank(A) ≤ min{m, n}. Using the above notations,<br />
the Moore-Penrose inverse of A has the following factorization<br />
A † = (I − K)F −1 V T . (3.10)<br />
Here I is the identity matrix of order n, K is a strict upper triangular matrix, F is a<br />
diagonal matrix whose diagonal elements are all positive.<br />
If A is full column rank, then<br />
V = A(I − K) (3.11)<br />
A † = (I − K)F −1 (I − K) T A T . (3.12)<br />
Proof<br />
Denote Āi = [a 1 , . . . , a i ]. Then since<br />
k i = A † i−1 a i (3.13)<br />
= [a 1 , . . . , a i−1 , 0, . . . , 0] † a i (3.14)<br />
= [ Ā i−1 , 0, . . . , 0 ] † ai (3.15)<br />
[ ]<br />
Ā†<br />
= i−1 a i (3.16)<br />
0<br />
⎡ ⎤<br />
k i,1<br />
.<br />
=<br />
k i,i−1<br />
0 , (3.17)<br />
⎢ ... ⎥<br />
⎣ ⎦<br />
0<br />
K = [k 1 , . . . , k n] is a strictly upper triangular matrix.<br />
Since u i = 0 ⇔ a i ∈ R(A i−1 ),<br />
{ ‖ui ‖ 2 2 if<br />
f i =<br />
1 + ‖k i ‖ 2 2 if<br />
a i ∉ R(A i−1 )<br />
a i ∈ R(A i−1 ) . (3.18)