Eigenvectors & Eigenvalues

Find the special directions that only stretch, never rotate. The key to understanding any linear transformation.

Eigenvectors: The Special Directions

Most vectors get rotated when you apply a matrix transformation. But eigenvectors are special—they only get stretched or flipped, never rotated.

The amount they stretch is called the eigenvalue. These special directions reveal the "natural axes" of any transformation.

The Core Idea

A·v = λ·v

When A transforms v, the result is just v scaled by λ. No rotation—just stretching!

  • v = eigenvector (the special direction)
  • λ = eigenvalue (the stretch factor)
  • λ > 0: same direction, λ < 0: flips direction

Demo 1: Hunt for Eigenvectors

Drag the blue vector around. When it aligns with an eigenvector direction, it turns green—the transformation only stretches it, without rotation.

Matrix A = [2, 1; 1, 2]
Eigenvector found!
This direction only stretches (by factor λ ≈ )
Eigenvector 1
λ = 3.00
Eigenvector 2
λ = 1.00

Drag the blue vector. When it turns green, you've found an eigenvector—a direction that only gets stretched, not rotated. The dashed lines show the actual eigenvector directions.

Demo 2: Eigenvalues in Action

Eigenvalues tell you how much an eigenvector stretches. Watch different matrices produce different eigenvalue behaviors.

A = [2, 0; 0, 3]→λ₁=2, λ₂=3 (both positive)
λ₁ = 3.00
Stretches
λ₂ = 2.00
Stretches

Eigenvalues tell you how much eigenvectors stretch. Positive λ means stretching, negative λ means flipping. Complex eigenvalues mean rotation without pure stretch directions.

Demo 3: Diagonalization (A = PDP⁻¹)

Any diagonalizable matrix can be decomposed into: change to eigenbasis, scale by eigenvalues, change back. This reveals the transformation's true nature.

A = PDP⁻¹
Where P = eigenvectors, D = diagonal eigenvalues

P⁻¹ transforms to the eigenbasis (eigenvectors become axes)

P
[-0.71, -0.71]
[-0.71, 0.71]
D
[3.00, 0]
[0, 1.00]
A
[2.00, 1.00]
[1.00, 2.00]

Diagonalization reveals that any matrix transformation can be understood as: change basis → scale along axes → change back. The eigenvectors define the natural axes for the transformation.

Demo 4: 3D Rotation Axes

In 3D, the rotation axis is the eigenvector with eigenvalue 1—the one direction that doesn't move at all during rotation.

Rotation Axis = Eigenvector with λ = 1
The one direction that doesn't move during rotation
Rotation around vertical axis
Axis: (0, 1, 0)→Eigenvector with λ = 1

Every 3D rotation has exactly one axis that stays fixed—the rotation axis. Vectors along this axis are eigenvectors with eigenvalue 1 (unchanged by rotation). The orange point traces its circular orbit while the purple axis remains still.

Key Takeaways

  • Eigenvectors are directions that only stretch (or flip), never rotate
  • Eigenvalues are the stretch factors (λ > 0 stretches, λ < 0 flips)
  • Complex eigenvalues mean the matrix rotates—no pure stretch directions exist
  • Diagonalization (A = PDP⁻¹) reveals any transformation as pure scaling in the right basis
  • Rotation axes in 3D are eigenvectors with λ = 1 (unchanged direction)

Next: Practice with the interactive sandbox and test your understanding!