[SOLVED] Finding the null space of a matrix

Finding the null space of a matrix

I'm trying to find the null space (solution space of Ax=0) of a given matrix. I've found two examples, but I can't seem to get either to work. Moreover, I can't understand what they're doing to get there, so I can't debug. Can someone walk me through this?

The documentation pages (numpy.linalg.svd, and numpy.compress) are opaque to me. I learned to do this by creating the matrix C = [A|0], finding the reduced row echelon form and solving for variables by row. I can't seem to follow how it's being done in these examples.

Thanks for any and all help!

Here is my sample matrix, which is the same as the wikipedia example:

A = matrix([
    [2,3,5],
    [-4,2,3]
    ])

Method (found here, and here):

import scipy
from scipy import linalg, matrix
def null(A, eps=1e-15):
    u, s, vh = scipy.linalg.svd(A)
    null_mask = (s <= eps)
    null_space = scipy.compress(null_mask, vh, axis=0)
    return scipy.transpose(null_space)

When I try it, I get back an empty matrix:

Python 2.6.6 (r266:84292, Sep 15 2010, 16:22:56) 
[GCC 4.4.5] on linux2
Type "help", "copyright", "credits" or "license" for more information.
>>> import scipy
>>> from scipy import linalg, matrix
>>> def null(A, eps=1e-15):
...    u, s, vh = scipy.linalg.svd(A)
...    null_mask = (s <= eps)
...    null_space = scipy.compress(null_mask, vh, axis=0)
...    return scipy.transpose(null_space)
... 
>>> A = matrix([
...     [2,3,5],
...     [-4,2,3]
...     ])  
>>> 
>>> null(A)
array([], shape=(3, 0), dtype=float64)
>>>

Solution

It appears to be working okay for me:

A = matrix([[2,3,5],[-4,2,3],[0,0,0]])
A * null(A)
>>> [[  4.02455846e-16]
>>>  [  1.94289029e-16]
>>>  [  0.00000000e+00]]