Given a matrix of vectors like:
$$ A = \begin{bmatrix} 1 & 5 \\ 2 & 4 \end{bmatrix} $$Subtract $\lambda$ along the diagonal, representing $A - \lambda I$: $$ A = \begin{bmatrix} 1 - \lambda& 5 \\ 2 & 4 - \lambda \end{bmatrix} $$
Find the determinant:
Now, finding the Eigenvectors, we know that $\lambda$ is either 6 or -1, so we plug these in:
$$ (A - \lambda I) = \begin{bmatrix} 1 & 5 \\ 2 & 4 \end{bmatrix} - \begin{bmatrix} 6 & 0 \\ 0 & 6 \end{bmatrix} \vec v = \begin{bmatrix} -5 & 5 \\ 2 & -2 \end{bmatrix} \vec v = \begin{bmatrix} 0 \\ 0 \end{bmatrix} $$Gives us $\vec v = \begin{bmatrix} 1 \\ 1 \end{bmatrix}$
Or, $$ (A - \lambda I) = \begin{bmatrix} 1 & 5 \\ 2 & 4 \end{bmatrix} - \begin{bmatrix} -1 & 0 \\ 0 & -1 \end{bmatrix} \vec v = \begin{bmatrix} 2 & 5 \\ 2 & 5 \end{bmatrix} \vec v = \begin{bmatrix} 0 \\ 0 \end{bmatrix} $$
Gives us $\vec v = \begin{bmatrix} 5 \\ -2 \end{bmatrix}$
ICA Question 2: Find the eigenvalues and the eigenvectors of the given matrix:
$$ A = \begin{bmatrix} -1 & 3 \\ 0 & 2 \end{bmatrix} $$import numpy as np
A = np.array([[-1, 3], [0, 2]])
# column 0 corresponds to eigenvalue 0
np.linalg.eig(A)
(array([-1., 2.]), array([[1. , 0.70710678], [0. , 0.70710678]]))