Math · Linear Algebra

Eigenvectors & Eigenvalues

Most vectors change direction under a transformation. Some special directions remain on the same line.

Concept

An eigenvector is a nonzero vector whose transformed image is a scalar multiple of itself: $A v = λ v$ .

Manipulate

Drag v, or focus its endpoint and use arrow keys. Find a direction where v and Av stay on the same line.

Observe

A positive eigenvalue preserves orientation; a negative one reverses it; zero collapses the vector. Rotation by 90° has no nonzero real eigenvector directions.

How do we find them?

$\det (A - λ I) = 0$ gives $λ^{2} - tr (A) λ + \det (A) = 0$ . The geometry comes first; this algebra finds the same special directions.

Computing, data & machine learning

Repeated transformations can emphasize directions associated with dominant eigenvalues. This is why eigenvectors help reveal structure in data, graphs, dynamics, and spectral methods.

Principal component analysis

If data forms a cloud of points, which directions capture its most important variation? PCA uses eigenvectors of a covariance relationship to answer that question.