Eigenvalues and Eigenvectors
What
For a square matrix A, an eigenvector v is a direction that doesn’t change when A is applied — it only gets scaled by a factor λ (the eigenvalue).
A @ v = λ × v
Why it matters
- PCA: principal components are eigenvectors of the covariance matrix — they point in the directions of maximum variance
- Spectral clustering: uses eigenvalues of the graph Laplacian
- Stability analysis: eigenvalues tell you if a system converges or diverges
- PageRank: the ranking vector is the dominant eigenvector of the link matrix
Key ideas
- Each square matrix has n eigenvalue-eigenvector pairs
- Eigenvectors are the “natural axes” of the transformation
- Eigenvalues tell you how much each axis stretches or shrinks
- Large eigenvalue = important direction, small = noise (PCA intuition)
In NumPy
import numpy as np
A = np.array([[4, 2], [1, 3]])
eigenvalues, eigenvectors = np.linalg.eig(A)
# eigenvalues: scaling factors
# eigenvectors: columns are the eigenvectors