本征分解让我纳闷 [英] Eigendecomposition makes me wonder in numpy

问题描述

我测试了一个定理，即A = Q * Lambda * Q_inverse，其中Q是具有特征向量的矩阵，而Lambda是具有对角特征值的对角矩阵.

I test the theorem that A = Q * Lambda * Q_inverse where Q the Matrix with the Eigenvectors and Lambda the Diagonal matrix having the Eigenvalues in the Diagonal.

我的代码如下:

import numpy as np
from numpy import linalg as lg

Eigenvalues, Eigenvectors = lg.eigh(np.array([

    [1, 3],

    [2, 5]


]))

Lambda = np.diag(Eigenvalues)


Eigenvectors @ Lambda @ lg.inv(Eigenvectors)

哪个返回:

array([[ 1.,  2.],
       [ 2.,  5.]])

返回的矩阵不应该与被分解的原始矩阵相同吗?

Shouldn't the returned Matrix be the same as the Original one that was decomposed?

推荐答案

您正在使用linalg.eigh函数，该函数用于对称/Hermitian矩阵，您的矩阵不是对称的.

You are using the function linalg.eigh which is for symmetric/Hermitian matricies, your matrix is not symmetric.

https://docs .scipy.org/doc/numpy-1.14.0/reference/generation/numpy.linalg.eigh.html

您需要使用linalg.eig，您将获得正确的结果:

You need to use linalg.eig and you will get the correct result:

https://docs.scipy.org /doc/numpy/reference/generated/numpy.linalg.eig.html

import numpy as np
from numpy import linalg as lg

Eigenvalues, Eigenvectors = lg.eig(np.array([

[1, 3],

[2, 5]


]))

Lambda = np.diag(Eigenvalues)


Eigenvectors @ Lambda @ lg.inv(Eigenvectors)

返回

[[ 1.  3.]
 [ 2.  5.]]

符合预期.

这篇关于本征分解让我纳闷的文章就介绍到这了，希望我们推荐的答案对大家有所帮助，也希望大家多多支持IT屋！