Help
What are Eigenvalues and Eigenvectors?
An eigenvector is a special vector that only gets scaled (not rotated) when a matrix transformation is applied to it.
Equation: Av = λv
- v is the eigenvector (direction that's preserved)
- λ (lambda) is the eigenvalue (how much it's scaled)
- A is the transformation matrix
Geometric Interpretation
For 2D transformations:
- Eigenvectors point in directions that don't rotate under the transformation
- Eigenvalues tell you how much the transformation stretches/compresses those directions
- Positive eigenvalue: vector maintains direction
- Negative eigenvalue: vector reverses direction
- Zero eigenvalue: dimension collapses (projection)
How to Find Eigenvalues
Solve the characteristic equation: det(A - λI) = 0
For a 2×2 matrix [a,b; c,d]:
- Subtract λ from diagonal: [a-λ,b; c,d-λ]
- Calculate determinant: (a-λ)(d-λ) - bc = 0
- Expand: λ² - (a+d)λ + (ad-bc) = 0
- Solve quadratic equation for λ
Matrix Properties
- Trace: Sum of diagonal elements = sum of eigenvalues
- Determinant: Product of eigenvalues
- Invertible: All eigenvalues must be non-zero
- Symmetric matrices: Always have real eigenvalues
Complex Eigenvalues
Real matrices can have complex eigenvalues (always in conjugate pairs).
Complex eigenvalues indicate rotational behavior:
- No real eigenvectors exist
- Common in rotation matrices
- Format: a ± bi
- Magnitude: √(a² + b²)
Applications
- PCA: Finding principal components in data
- Physics: Quantum mechanics, vibration analysis
- Graphics: 3D transformations
- PageRank: Web page ranking algorithm
- Stability: Analyzing system behavior over time
Understanding the Examples
- Identity: All eigenvalues = 1 (no transformation)
- Diagonal: Eigenvalues are the diagonal entries
- Rotation: Complex eigenvalues (unless 180°)
- Scaling: Eigenvalues = scale factors
- Reflection: Eigenvalues are +1 and -1
- Shear: One eigenvalue = 1, eigenvector along shear axis