如何用NumPy解齐线性方程组? [英] How to solve homogeneous linear equations with NumPy?

问题描述

如果我有这样的齐次线性方程式

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.  ]])

我想为此提供一个非零的解决方案.如何使用 NumPy 完成?

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

编辑

linalg.solve仅适用于A * x = b，其中b不仅包含0.

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

推荐答案

您可以使用SVD或QR分解来计算线性系统的空空间，例如:

You can use an SVD or a QR decomposition to compute the null space of the linear system, e.g., something like:

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

以您的示例为例:

>>> 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]])

另请参见> 空空间的数值计算 在Wikipedia上.

See also the section Numerical computation of null space on Wikipedia.

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