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 knocked off their line by a matrix. Eigenvectors don't: they only stretch or flip, and the stretch factor is the eigenvalue.

Hunt for Eigenvectors

Drag the blue vector around. When it turns green you've found an eigenvector: 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.

Go deeper:The Core Idea

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.

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

Eigenvalues in Action

Pick a preset matrix and watch each eigenvector grow, shrink, or flip by its λ. Try Rotation 90°: it has no real eigenvectors at all.

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.

Diagonalization (A = PDP⁻¹)

Step through, or press Auto-Play, to see A as three moves: change to the eigenbasis (P⁻¹), scale by the eigenvalues (D), and change back (P).

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

Go deeper: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.

3D Rotation Axes

Choose a rotation axis and watch everything spin around it. The axis is the eigenvector with eigenvalue 1: the one direction that doesn't move.

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

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