Eigen values and vectors calculation error in python

Question

I'm trying to obtain the eigenvectors and values of any matrix 'X' in a specific format. I used the linalg function to get the eigen pairs but the expected output format is different from my result. For example, v and e denote the eigenvalues and eigenvectors. v1 = 1, e1 = [1,0,0], v2 = 2, e2 = [0,1,0], v3 = 3, e3 = [0,0,1].

So in this example, the eigen pairs of matrix X should be Ep =[(1, [1,0,0]) (2, [0,1,0]), (3, [0,0,1])]. Here P[0] represents the first eigen pair (1,[1,0,0]), where the eigenvalue is 1, and the eigenvector is [1,0,0].

Can you please help me code this part further?

e,v = np.linalg.eigh(X)

Answer 1

np.linalg.eigh

First, one should note that np.linalg.eigh calculates the eigenvalues of a Hermitian matrix -- this will not apply for all matrices. If you want to calculate the eigenvalues of any matrix X you should probably switch to something like np.linalg.eig:

import numpy as np

L = np.diag([1,2,3])
V = np.vstack(([1,0,0],[0,1,0],[0,0,1]))

# X = V@L@V.T (eigendecomposition)
X = V@L@V.T

w,v = np.linalg.eig(X)
assert (np.diag(w) == L).all()
assert (v == V).all()

Eigenpairs

To construct the eigenpairs, just use some list comprehension:

import numpy as np

# X = V@L@V.T (eigendecomposition)
X = np.diag([1,2,3])

w,v = np.linalg.eig(X)

Ep = [(val,vec.tolist()) for val,vec in zip(w,v)]

Enjoy!