Finding Eigenvalues and Eigenvectors Guide | Generated by AI
Eigenvalues
To find eigenvalues of a matrix A, you solve the characteristic equation:
det(A - λI) = 0
where λ (lambda) represents the eigenvalues and I is the identity matrix.
Steps:
- Subtract λ from the diagonal elements of A to get (A - λI)
- Calculate the determinant of (A - λI)
- Set the determinant equal to zero
- Solve the resulting polynomial equation for λ
Eigenvectors
Once you have an eigenvalue λ, find its corresponding eigenvector by solving:
(A - λI)v = 0
where v is the eigenvector (a non-zero vector).
Steps:
- Substitute the eigenvalue λ into (A - λI)
- Solve the homogeneous system of linear equations
- Find all non-zero vectors that satisfy the equation
- The solution space forms the eigenspace for that eigenvalue
Example
For a 2×2 matrix A = [[3, 1], [1, 3]]:
Finding eigenvalues:
- A - λI = [[3-λ, 1], [1, 3-λ]]
- det(A - λI) = (3-λ)² - 1 = λ² - 6λ + 8 = 0
- Solving: λ = 4 and λ = 2
Finding eigenvector for λ = 4:
- (A - 4I)v = 0 gives [[−1, 1], [1, −1]]v = 0
- This yields v = [[1], [1]] (or any scalar multiple)
Finding eigenvector for λ = 2:
- (A - 2I)v = 0 gives [[1, 1], [1, 1]]v = 0
- This yields v = [[1], [−1]]
For numerical computation or larger matrices, software like Python (NumPy), MATLAB, or other linear algebra tools can compute these automatically.