Eigenvalue and Eigenvector Calculator
Calculate eigenvalues and eigenvectors for square matrices (2×2 and 3×3).
Calculate Eigenvalues and Eigenvectors
Eigenvalue Examples
2×2 Matrix
Matrix: 2,1;1,2
Eigenvalues: 3, 1
Eigenvectors: [1,1], [1,-1]
Diagonal Matrix
Matrix: 3,0;0,5
Eigenvalues: 3, 5
Eigenvectors: [1,0], [0,1]
Symmetric Matrix
Matrix: 1,2;2,1
Eigenvalues: 3, -1
Eigenvectors: [1,1], [1,-1]
How to Calculate Eigenvalues and Eigenvectors
Definition
For matrix A, eigenvalue λ and eigenvector v satisfy: Av = λv
Characteristic Polynomial
Solve: det(A - λI) = 0
Where I is the identity matrix.
2×2 Matrix
For matrix [[a,b],[c,d]]:
λ² - (a+d)λ + (ad-bc) = 0
3×3 Matrix
For 3×3 matrix, solve cubic equation:
λ³ - trace(A)λ² + sum(minors)λ - det(A) = 0
Finding Eigenvectors
For each eigenvalue λ, solve: (A - λI)v = 0
Properties
- Sum of eigenvalues = trace of matrix
- Product of eigenvalues = determinant
- Eigenvectors are linearly independent
- Real symmetric matrices have real eigenvalues