What is the geometric meaning of eigenvalues and eigenvectors?
Updated May 16, 2026
Short answer
Eigenvectors are directions unchanged by a transformation, and eigenvalues scale them.
Deep explanation
A matrix transforms space by rotating, stretching, or shearing it. Eigenvectors represent special directions that remain on their own span after transformation. Eigenvalues indicate how much those directions are stretched or compressed. This reduces complex transformations into simple scalar scaling along key axes.
Real-world example
Used in PCA to find directions of maximum variance in data.
Common mistakes
- Thinking every matrix has real eigenvectors (not true for all cases).
Follow-up questions
- What does a negative eigenvalue mean?
- Why are eigenvectors important in ML?