Find the eigenvalues and corresponding eigenvectors of the matrix:
0 0 1
M = 1 0 0
0 1 0
L1 = 1
L2 = -.5 - .866 i
L3 = -.5 + .866 i
eigenvectors messy
see
http://www.wolframalpha.com/widgets/view.jsp?id=9aa01caf50c9307e9dabe159c9068c41
.866 is sin 60 by the way
To find the eigenvalues and corresponding eigenvectors of a matrix, you need to solve the characteristic equation and then find the corresponding eigenvectors.
First, let's find the eigenvalues by solving the characteristic equation:
The characteristic equation is given by det(M - λI) = 0, where M is the matrix, λ is the eigenvalue, and I is the identity matrix.
Substituting the given matrix, we have:
det(M - λI) = det([[0-λ, 0, 1], [1, 0-λ, 0], [0, 1, 0-λ]])
Expanding the determinant, we get:
det([[0-λ, 0, 1], [1, 0-λ, 0], [0, 1, 0-λ]]) = (0-λ) * (0-λ)(0-λ) - 1 * (0-λ) *1 - 0 * (0-λ) * 0 - (0-λ) * (1 * (0-λ) - 0 * 1)
Simplifying this expression, we have:
(0-λ) * (0-λ)(0-λ) - (0-λ)^2 = -λ(λ^2 - λ - 1) + (λ^2 - 2λ + 1) = -λ^3 + 2λ^2 - 2λ + 1
To find the eigenvalues, we set -λ^3 + 2λ^2 - 2λ + 1 = 0 and solve for λ.
Solving this equation is difficult analytically, but we can use numerical methods to find the approximate values of the eigenvalues. One such method is the Newton-Raphson Iteration or the Newton's method.
Alternatively, you can use math software like MATLAB or Python with libraries such as NumPy to calculate the eigenvalues numerically. Here's an example using Python:
import numpy as np
M = np.array([[0, 0, 1], [1, 0, 0], [0, 1, 0]])
eigenvalues, eigenvectors = np.linalg.eig(M)
print("Eigenvalues:", eigenvalues)
The output will be the array of eigenvalues.
To find the corresponding eigenvectors, you substitute each eigenvalue back into the equation (M - λI)x = 0 and solve for x.
Using the same example with Python, we can find the corresponding eigenvectors:
import numpy as np
M = np.array([[0, 0, 1], [1, 0, 0], [0, 1, 0]])
eigenvalues, eigenvectors = np.linalg.eig(M)
print("Eigenvalues:", eigenvalues)
print("Eigenvectors:", eigenvectors)
The output will be the array of eigenvalues and the matrix of eigenvectors, where each column represents an eigenvector.