Lesson 6 of 10Article20 min
Eigenvalues & Eigenvectors
Av = λv with v ≠ 0 means A stretches v by factor λ. Eigenvectors reveal the natural axes of a linear transformation.
Directions that only stretch
Av = λv with v ≠ 0 means A stretches v by factor λ. Eigenvectors reveal the natural axes of a linear transformation.
- Trace = sum of eigenvalues; det = product (over ℂ, counting multiplicity).
- Repeated eigenvalues may fail to yield enough independent eigenvectors.
- Powers: Aᵏ = PDᵏP⁻¹ when diagonalisable — useful for recurrence relations.
In data science, PCA finds directions of maximum variance — eigenvectors of a covariance matrix.