Re: singular value decomposition & zero singular values


Steve Hawkwind wrote:
I am using Algorithm 8.6.2 of Golub and Van Loan to find the SVD of the
following matrix :-


2 2 -2
A = 2 2 -2
-2 -2 6


This has singular values of 8,2, and 0

I have bi-diagonalized the matrix and then applied the algorithm to it and
for every iteration of the algorithm observing the resulting effect. It
seems that the '8' converges quite quickly but the '2' converges on the
sub-diagonal. Heres a sample run :-

First iteration
7.9769458476 0.6026593095 0
0 0.9211435111 -1.7766911191
0 0 0



Second iteration
7.9999161027 0.0366379321 0
0 0.0222258777 -1.9998765015
0 0 0




Third iteration
7.9999996272 0.0024424036 0
0 -1.0E-9 2.0000000000
0 0 0




Fourth iteration
7.9999999983 0.0001628269 0
0 0 2.0000000000
0 0 0


Now according to G&VL, if a zero appears on the central diagonal (it has, at
position 2,2) then the corresponding superdiagonal entry should be made
zero. But this would blank out the '2' which is the other singular value
which is needed? Surely this destroys information.

I've tried other matrices containing zero singular values and the same story
there, the remaining singular values appear on the superdiagonal and
according to G&VL they should be zeroed...

Is there a mistake here?


Thanks
Hi,
yes, there is a mistake here. It is your misunderstanding of the word
"to zero" an element. This does not mean to set it to zero but to apply
a similarity transformation (here a Givens rotation) to make it zero.

.

Relevant Pages