Eigenvectors' corresponding eugenvalues in Python

Question

I have a symmetric 3*3 matrix, A:

A = np.mat("1, 2, 1; 2, 5, 0; 1, 0, 5")

I have calculated the eigenvalues of A, which is 0, 6 and 5. I have now been told that A has the following three eigenvectors, and I am asked to find the corresponding three eigenvalues. How do I do this in Python?

   v1=np.array([[-5],[2], [1]])
   v2=np.array([[0],[-1], [2]])
   v3=np.array([[1],[2], [1]])

Afterwards I have to show that the three eigenvectors are linearly independent, which I also don't know how to show in Python?

Answer 1

To find the eigenvalue of a given eigenvector there is a well known formula (Rayleigh quotient):

And to find if the eigenvectors are linearly independent, you can show that their determinant is different than zero:

import numpy as np

A = np.array([[1, 2, 1], [2, 5, 0], [1, 0, 5]])

v1 = np.array([[-5], [2], [1]])
v2 = np.array([[0], [-1], [2]])
v3 = np.array([[1], [2], [1]])

# eigenvalues
l1 = (v1.T@A@v1/(v1.T@v1))[0, 0]
l2 = (v2.T@A@v2/(v2.T@v2))[0, 0]
l3 = (v3.T@A@v3/(v3.T@v3))[0, 0]

# Are linearly independent?
tol = 1e-8
if np.abs(np.linalg.det(np.concatenate((v1, v2, v3), axis=1))) < tol:
    print('eigenvector are linearly dependent')
else:
    print('eigenvector are linearly independent')