Eigenvectors: Definition and Transformation Examples

Directions that remain on the same line under a linear transformation

A nonzero vector x⃗ is an eigenvector of a matrix A when multiplication by A leaves it on the same line. The scalar λ is the corresponding eigenvalue.

Explore more on GraphMath Linear Algebra

Definition and examples

Eigenvectors definition and examples table comparing common 2D linear transformations, their eigenvalues and the directions preserved by each matrix.
The table compares common 2D transformations and shows whether every direction, selected directions or no real direction satisfies A x⃗ = λx⃗. Click the image to open the full-size version.

Concept

The equation A x⃗ = λx⃗ says that the transformation acts on the eigenvector direction as a one-dimensional scaling. The vector may be stretched, shrunk, reversed or collapsed, but it is not turned away from its original line.

Structure

Uniform scaling preserves every direction. Nonuniform diagonal scaling preserves the coordinate axes. A shear typically preserves one direction. A real symmetric matrix has orthogonal eigenvector directions. Projection preserves one subspace and collapses its orthogonal complement, while reflection preserves one direction and reverses the perpendicular one.

A genuine 2D rotation generally has no real eigenvectors because every nonzero real vector changes direction.

Key equations

A x⃗ = λx⃗, x⃗ ≠ 0⃗
λ > 1: stretch
0 < λ < 1: shrink
λ < 0: reverse direction
λ = 0: collapse to 0⃗

Query phrases

  • eigenvector definition visualized
  • geometric meaning of eigenvectors
  • eigenvectors of common transformations
  • eigenvectors of scaling shear projection reflection rotation
  • what does A x equals lambda x mean
  • why rotation has no real eigenvectors
  • eigenvalue geometric interpretation