IX · Linear algebra
Eigenvalues & eigenvectors
What it is
For matrix A, vector v with Av = λv. Captures invariant directions of a linear transformation. Power iteration finds the dominant one.
Where it lives
PageRank (the dominant eigenvector of the link matrix), spectral clustering, PCA, stability of dynamical systems.
The key insight
A matrix's eigenvalues encode its long-run behaviour: |λ| > 1 means amplification, |λ| < 1 means decay, complex λ means rotation.