SVD question

[smritibhagat]

Hi!
I have been trying to figure this out, and need help...
How do I compute an orthogonal complement of a matrix using SVD?
Is there a python lib function or code that does this?
Thanks!

 

[Robert Kern]

Hi!
I have been trying to figure this out, and need help...
How do I compute an orthogonal complement of a matrix using SVD?

On the chance that this is homework, I will only point out that Golub and van
Loan's book _Matrix Computations_ is essential reading if you are doing, well,
matrix computations.

Is there a python lib function or code that does this?

numpy has SVD.

http://numeric.scipy.org

--
Robert Kern
(e-mail address removed)

"I have come to believe that the whole world is an enigma, a harmless enigma
that is made terrible by our own mad attempt to interpret it as though it had
an underlying truth."
-- Umberto Eco

 

[smritibhagat]

Hi Robert!
Oh! Its not a homework problem...
I read the Golub book, it tells me what an orthogonal complement is,
however, I cannot understand how I can code it.
I know about svd from numpy's mlab, but I what I want to know is how
can I compute an orthogonal complement, using SVD or otherwise.
Thanks for the prompt reply

 

[Robert Kern]

Hi Robert!
Oh! Its not a homework problem...
I read the Golub book, it tells me what an orthogonal complement is,
however, I cannot understand how I can code it.
I know about svd from numpy's mlab, but I what I want to know is how
can I compute an orthogonal complement, using SVD or otherwise.

Assuming A is an array with the vectors as columns and has shape (m, n), then
the null space of A (= the orthogonal complement of the vectors assuming that
the set of vectors is linearly independent):

In [231]: A
Out[231]:
array([[ 0., 1.],
[ 1., 1.],
[ 2., 1.],
[ 3., 1.]])

In [232]: m, n = A.shape

In [233]: u, s, vh = numpy.linalg.svd(A)

In [234]: dot(transpose(u[:, n:]), A)
Out[234]:
array([[ 0.00000000e+00, -1.11022302e-16],
[ -1.42247325e-16, -5.65519853e-16]])

In [235]: ortho_complement = u[:, n:]

In [236]: ortho_complement
Out[236]:
array([[-0.38578674, -0.38880405],
[ 0.22458489, 0.80595386],
[ 0.70819044, -0.44549557],
[-0.54698859, 0.02834576]])

--
Robert Kern
(e-mail address removed)

"I have come to believe that the whole world is an enigma, a harmless enigma
that is made terrible by our own mad attempt to interpret it as though it had
an underlying truth."
-- Umberto Eco

 

[smritibhagat]

Thanks Robert!
I was using mlab's svd function, which returns an mxn matrix for u, and
hence was unable to see how to compute the orthogonal complement!
I realize that numpy's svd gives the mxm matrix!
Thanks again.
-Smriti