Lesson 4.1: Eigenvectors & Eigenvalues

Discovering the Soul of a Matrix

We've seen that a matrix is a transformation that can stretch, shrink, shear, and rotate space. Most vectors, when transformed, are knocked off their original path. But what if there are special vectors? Vectors that, when transformed, are only stretched or shrunk, but not rotated?

These special, non-rotating vectors are the "invariant axes" of a transformation. They are the eigenvectors of a matrix.

The corresponding factor by which they are stretched or shrunk is their eigenvalue. Understanding these concepts is the key to unlocking the deep structure of a matrix and is the foundation for countless applications, from Principal Component Analysis (PCA) to solving differential equations.

The Mathematical Definition

The Eigen-Equation

An eigenvector `v` and its corresponding eigenvalue `λ` for a matrix `A` are a pair that satisfies the following equation:

Av=λvAv = \lambda v
  • `A` is our `n x n` transformation matrix.
  • `v` is the `n x 1` eigenvector (it's a non-zero vector).
  • `λ` (lambda) is the scalar eigenvalue.

This equation perfectly captures the concept: "The action of matrix `A` on the vector `v` is the same as just scaling `v` by the number `λ`."

Try The Interactive Visualizer
The best way to understand this concept is to see it. Our interactive tool in the QuantLab lets you apply transformations and see the eigenvectors in action.
Go to Eigen-Visualizer

How to Find Them: The Characteristic Equation

To find these special pairs, we need to do a little algebraic manipulation.

Av−λv=0Av - \lambda v = 0

To factor out `v`, we need to introduce the identity matrix `I`:

(A−λI)v=0(A - \lambda I)v = 0

This gives us our master key. To find the eigenvalues, we must solve the characteristic equation:

det⁡(A−λI)=0\det(A - \lambda I) = 0

Solving this equation for `λ` will give us the eigenvalues. Then, for each `λ`, we can plug it back into `(A - λI)v = 0` to find the corresponding eigenvectors `v` in the null space.

Summary: The Soul of a Matrix
  • Eigenvectors are the vectors that are only scaled (not rotated) by a transformation. They define the transformation's invariant axes.
  • Eigenvalues are the scalar factors by which their corresponding eigenvectors are scaled.
  • The defining relationship is `Av = λv`.
  • We find eigenvalues by solving the characteristic equation: `det(A - λI) = 0`.
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Lesson 3.2: Calculation and Properties
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Lesson 4.2: The Characteristic Equation