Eigenvalues & Eigenvectors: The Axis of Rotation

1. Introduction: The Spinning Earth

Imagine the Earth spinning on its axis.

  • Someone standing in Brazil is moving very fast (around the center of the Earth).
  • Someone standing in London is moving, but slower.
  • But someone standing exactly on the North Pole is not moving at all (relative to the axis). They are just spinning in place.

In Linear Algebra, the North Pole-South Pole line is the Eigenvector of the Earth’s rotation.

When a matrix A transforms a vector v, the result Av usually points in a completely new direction. But for Eigenvectors, the result points in the same direction (or directly opposite). It only gets stretched or shrunk.

The Equation

Av = λv
  • A: The Transformation Matrix (The “Action”).
  • v: The Eigenvector (The “Direction”).
  • λ (Lambda): The Eigenvalue (The “Stretch Factor”).

2. Interactive Visualizer: The Eigen-Spinner v4.0

Below is a 2D space. The matrix A transforms the Blue Vector (v) into the Green Vector (Av). Your Goal: Find the Eigenvectors by rotating v until it aligns with Av.

  • Blue Arrow: Input Vector v.
  • Green Arrow: Output Vector Av.
  • Red Glow: Indicates you found an Eigenvector!
A =
Determinant (Area): 3.00
Trace (Sum Diag): 4.00
Input v: [1.0, 0.0]
Output Av: [2.0, 1.0]
EIGENVECTOR FOUND! λ = ?

3. Computing Eigenvalues (The Math)

How do we find λ without guessing? We want to solve Av = λv.

  1. Move everything to one side:
    Av - λIv = 0
    (A - λI)v = 0
  2. Geometric Intuition: For a non-zero solution v to exist, the matrix (A - λI) must squash space into a lower dimension (like squashing a 2D plane into a line).
    • This means the Area (Determinant) must be zero.
  3. Therefore, we solve:
det(A - λI) = 0

This is called the Characteristic Equation.

Example: The [2, 1], [1, 2] Matrix

Let A =

21
12

.

  1. Subtract λ from the diagonal:
    2 - λ1
    12 - λ
  2. Find the Determinant (ad - bc):
    (2 - λ)(2 - λ) - (1)(1) = 0
    (4 - 4λ + λ2) - 1 = 0
    λ2 - 4λ + 3 = 0
  3. Solve the Quadratic Equation:
    (λ - 3)(λ - 1) = 0

So, the Eigenvalues are λ = 3 and λ = 1. (Go back to the visualizer and set the angle to 45° or 135° to see these!)

4. Application: Eigenfaces (Face Recognition)

Before Deep Learning, Face Recognition used Eigenvalues.

  1. Imagine a face image is just a vector of pixels (e.g., 10,000 pixels = a 10,000D vector).
  2. If you take photos of 1,000 people, you have a cloud of points in 10,000D space.
  3. We calculate the Covariance Matrix of these faces.
  4. The Eigenvectors of this matrix are called Eigenfaces.

Why? Because they represent the “Principal Components” of human faces—the basic ingredients that make up a face (e.g., nose shape, eye distance). Any face can be reconstructed by adding up a few weighted Eigenfaces!

5. Summary

  • Eigenvector: An axis that doesn’t rotate, only stretches.
  • Eigenvalue: The amount of stretch along that axis.
  • Characteristic Equation: det(A - λI) = 0.
  • Trace: Sum of eigenvalues.
  • Determinant: Product of eigenvalues.

Next: Matrix Decompositions (SVD) →