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. I'm hoping someone might be able to 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)
>>>
Sympy makes this straightforward.
>>> from sympy import Matrix
>>> A = [[2, 3, 5], [-4, 2, 3], [0, 0, 0]]
>>> A = Matrix(A)
>>> A * A.nullspace()[0]
Matrix([
[0],
[0],
[0]])
>>> A.nullspace()
[Matrix([
[-1/16],
[-13/8],
[ 1]])]