Eigenvalues & Eigenvectors
Vectors that don't change direction under transformation.
λThe Eigen Equation
Av = λv
v (Eigenvector)
A non-zero vector that only gets scaled by the linear transformation A. It does not change direction.
λ (Eigenvalue)
The scalar factor by which the eigenvector is stretched or shrunk.
How to Find Them
To find the eigenvalues, we solve the characteristic equation:
det(A - λI) = 0
Once λ is found, substitute it back into (A - λI)v = 0 to solve for the eigenvector v.
Applications
- •Google PageRank: Uses eigenvectors of the web graph.
- •Vibration Analysis: Natural frequencies of bridges/buildings.
- •Face Recognition: Eigenfaces in computer vision.
- •Quantum Mechanics: States and observables.