How to solve homogeneous linear equations with NumPy?

Question

If I have homogeneous linear equations like this

array([[-0.75,  0.25,  0.25,  0.25],
       [ 1.  , -1.  ,  0.  ,  0.  ],
       [ 1.  ,  0.  , -1.  ,  0.  ],
       [ 1.  ,  0.  ,  0.  , -1.  ]])

And I want to get a non-zero solution for it. How can this be done with NumPy ?

EDIT

linalg.solve only works on A * x = b, where b does not contain only 0.

Answer 1

You can use SVD or QR decomposition to calculate the zero space of a linear system, for example:

import numpy

def null(A, eps=1e-15):
    u, s, vh = numpy.linalg.svd(A)
    null_space = numpy.compress(s <= eps, vh, axis=0)
    return null_space.T

This gives for your example:

>>> A
matrix([[-0.75,  0.25,  0.25,  0.25],
        [ 1.  , -1.  ,  0.  ,  0.  ],
        [ 1.  ,  0.  , -1.  ,  0.  ],
        [ 1.  ,  0.  ,  0.  , -1.  ]])

>>> null(A).T
array([[-0.5, -0.5, -0.5, -0.5]])

>>> (A*null(A)).T
matrix([[ 1.66533454e-16, -1.66533454e-16, -2.22044605e-16, -2.22044605e-16]])

See also the section Numerical Computation of Zero Space on Wikipedia.

Answer 2

, . , U , :

import numpy as np

def solution(U):
    # find the eigenvalues and eigenvector of U(transpose).U
    e_vals, e_vecs = np.linalg.eig(np.dot(U.T, U))  
    # extract the eigenvector (column) associated with the minimum eigenvalue
    return e_vecs[:, np.argmin(e_vals)]