Eigenvalue and Eigenvector Calculator
Calculate eigenvalues and eigenvectors for 2×2 and 3×3 matrices with step-by-step solutions
Matrix Input
Enter real numbers for matrix elements
Results
Eigenvalues
Eigenvectors
Characteristic Polynomial
det(A - λI) = λ² - 5.000λ + -2.000 = 0
Formulas Used
For 2×2 matrices:
λ = (tr(A) ± √(tr(A)² - 4|A|)) / 2
Eigenvector equation: (A - λI)v = 0
Trace: Sum of diagonal elements
Determinant (2×2): ad - bc
Example Calculation
2×2 Matrix Example
Matrix A: [[3, 1], [0, 2]]
Trace: 3 + 2 = 5
Determinant: 3×2 - 1×0 = 6
Characteristic equation: λ² - 5λ + 6 = 0
Solution
Eigenvalues: λ₁ = 3, λ₂ = 2
Eigenvector for λ₁ = 3: [1, 0]
Eigenvector for λ₂ = 2: [-1, 1]
Key Concepts
Eigenvalue
Scalar that scales eigenvector
Eigenvector
Vector that doesn't change direction
Trace
Sum of diagonal elements
Applications
Principal Component Analysis (PCA)
Stability analysis of systems
Google PageRank algorithm
Quantum mechanics
Machine learning algorithms
Understanding Eigenvalues and Eigenvectors
What are Eigenvalues and Eigenvectors?
An eigenvector of a square matrix A is a non-zero vector v that, when A is multiplied by v, yields a scalar multiple of v. The scalar multiple is called the eigenvalue λ.
A × v = λ × v
Characteristic Equation
To find eigenvalues, we solve the characteristic equation:
det(A - λI) = 0
For 2×2 Matrices
For a 2×2 matrix, eigenvalues can be found using the quadratic formula:
λ = (tr(A) ± √(tr(A)² - 4|A|)) / 2
Properties
- • Sum of eigenvalues equals the trace
- • Product of eigenvalues equals the determinant
- • Eigenvectors are linearly independent
- • Complex eigenvalues occur in conjugate pairs
Note: For 3×3 matrices, finding exact eigenvalues requires solving cubic equations, which can be computationally intensive.