Big idea
A matrix usually turns a vector. But an eigenvector keeps its direction:
A v = λ v
The number λ is the stretch factor. Move through the tools below to see this idea from geometry, dynamics, ranking, and data.
1. Matrix action and surviving directions
Choose a matrix. The thin circle shows input directions. The transformed curve shows where the matrix sends them. Eigenvectors are directions that land on the same line.
2. Test one vector
Move the vector and compare x with Ax. When they are on the same line, the vector is in an eigen-direction.
3. Repeated matrix action
Repeated multiplication often pulls a vector toward the dominant eigen-direction.
If the largest eigenvalue in absolute value is clearly bigger than the others, its eigenvector usually dominates long-term behavior.
4. Ranking as an eigenvector
A ranking vector is stable when one update only scales it or leaves it unchanged. This is the eigenvector idea in network form.
5. Data variation and eigenvectors
Eigenvectors of a covariance matrix point in directions where data varies. This is the beginning of PCA.
Reflection questions
- How can you recognize an eigenvector from a picture?
- What does a negative eigenvalue do geometrically?
- Why does a zero eigenvalue mean information loss?
- Why do eigenvectors matter for repeated matrix action?
- How do eigenvectors connect to PCA and data variation?