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.  ],
          


        

Answer 1

For that matter, the best solution of an over constrained homogeneous linear system is the eigenvector associated with the smallest eigenvalue. So given U as the coefficient matrix of the system, the solution is:

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)]