Geometric Interpretation

Linear Algebra for Machine Learning · Axiom Academy

Visualizing how eigenvectors behave under transformations When a matrix represents a rotation, it doesn't have real eigenvectors (unless it's a 180° rotation). The rotation transforms all vectors without preserving direction. Red vector rotating; no eigenvectors for typical rotations A scaling matrix multiplies each coordinate by a constant. Every vector is an eigenvector! All vectors stretch along their original directions A shear matrix has eigenvectors that point along specific directions. One eigenvector typically stays fixed, while others are scaled. Green vector (eigenvector) stays on its line; red vector rotates The green vector doesn't change direction—it's an eigenvector with eigenvalue 1. The red vector's direction changes—not an eigenvector. This is why eigenvectors are so important in applications: Principal Component Analysis (PCA): The eigenvectors of the covariance matrix are the principal axes of your data. Stability Analysis: Eigenvectors describe the modes of a system; eigenvalues tell you their growth/decay rates. PageRank: The eigenvector represents the stationary distribution of web visits.

This is the written version of the interactive lesson above. See the full Linear Algebra for Machine Learning course.