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.
My code is the following:
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)
Which returns :
array([[ 1., 2.],
[ 2., 5.]])
Shouldn't the returned Matrix be the same as the Original one that was decomposed?
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/generated/numpy.linalg.eigh.html
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)
returns
[[ 1. 3.]
[ 2. 5.]]
As expected.