Module 6 · Lesson 6.5

Eigenvectors & Eigenvalues — Directions That Don't Turn

Find the special directions a matrix only stretches or shrinks, never rotates — and see why those directions matter for describing data.

Intuition

Pick almost any vector and apply a matrix to it, and it gets pushed off in a brand new direction — rotated as well as resized. But for a handful of special directions, something simpler happens: the matrix only stretches or shrinks the vector, leaving its direction completely untouched. Those special directions are the eigenvectors, and how much they stretch is the eigenvalue.

Core concept

A nonzero vector $\vec v$ is an eigenvector of $A$ if $A\vec v$ comes out as a plain scalar multiple of $\vec v$ itself — same direction (or exactly reversed), just longer or shorter. That scalar is the eigenvalue $\lambda$. Most matrices have only a couple of these special directions, buried among infinitely many ordinary ones that do get rotated.

v and Av, same direction
Matrix A[[2, 1], [1, 2]]
Test vector v(1, 1)
Compute Av(2·1+1·1, 1·1+2·1) = (3, 3)
Compare to v(3, 3) = 3 × (1, 1)
Answerv=(1,1) is an eigenvector, λ = 3

Interactive Lab

Slide to sweep a test vector $\vec v$ (dashed) through every direction. Watch its image $A\vec v$ (solid) — most of the time it points somewhere else entirely. At two special angles, it snaps into perfect alignment with $\vec v$.

θ = 0° angle between v and Av = 45.0° status: rotated

Key Idea

$$ A\vec{v} = \lambda \vec{v} $$

This single equation defines everything: apply the matrix, and instead of landing somewhere new, the eigenvector $\vec v$ just gets rescaled by the eigenvalue $\lambda$. Every other direction gets bent away from itself; these directions don't.

$A$The transformation matrix being applied
$\vec{v}$An eigenvector — its direction is preserved by $A$
$\lambda$The eigenvalue — the factor $\vec v$ is stretched or shrunk by
$A\vec{v}$The transformed vector, exactly parallel to $\vec v$

Think Further

  1. PCA finds the eigenvectors of a dataset's covariance matrix and treats them as the data's "principal directions." Why would directions a matrix only stretches, without rotating, be a natural choice for describing a dataset's most important directions?
  2. If a matrix has two eigenvalues like 3 and 1 here, what happens to a generic vector's direction after the matrix is applied to it many times in a row? Which eigen-direction would you expect it to drift toward?
  3. Predict what an eigenvalue of exactly 0 would mean about that eigenvector's direction, connecting back to the null space idea from Lesson 6.3.
Show suggested answers
  1. Because eigenvectors are the directions where the transformation's effect is purest — a clean stretch with no rotation muddying the picture — so they capture the axes along which the data spreads out most distinctly, without directional distortion mixing separate axes together.
  2. The component along the eigenvalue-3 direction keeps tripling with each application while the eigenvalue-1 component stays constant, so the faster-growing direction increasingly dominates — the vector's direction drifts closer and closer to the eigenvalue-3 eigenvector over repeated applications.
  3. An eigenvalue of 0 means $A\vec v = 0\cdot\vec v = \vec 0$ — that eigenvector's direction is exactly the collapsed direction from Lesson 6.3's null space, since the transformation squashes it all the way down to the zero vector.