Eigenvalues & Eigenvectors

Most matrices, when they act on a vector, both rotate it and scale it. But for every square matrix there are special directions where the matrix only scales the vector — it doesn't rotate it at all. Those directions are the eigenvectors, and the amount they get scaled by is the eigenvalue.

The definition

For a square matrix $A$ , a nonzero vector $\vec{v}$ is an eigenvector if:

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

where $\lambda$ (lambda) is a scalar — the eigenvalue. The equation says: applying $A$ to $\vec{v}$ gives you back the same vector, just stretched or shrunk by $\lambda$ .

Watch it

Manim animation placeholder — render your scene with manim -qh scene.py SceneName and drop the .mp4 into /public/videos/, then set src here.

A Manim animation showing a matrix transforming the plane, with eigenvectors highlighted as the directions that don't rotate.

To produce this clip yourself with the ManimCommunity library:

from manim import *

class EigenScene(Scene):
    def construct(self):
        plane = NumberPlane()
        matrix = [[2, 1], [1, 2]]
        self.add(plane)
        self.play(plane.animate.apply_matrix(matrix), run_time=3)

Why it matters

Eigenvalues show up everywhere: Google's PageRank, vibration modes in engineering, principal component analysis in data science, and quantum mechanics all boil down to finding eigenvalues of some matrix.

Try it yourself

\det(A - \lambda I) = 0

Solving this "characteristic equation" is how you find the eigenvalues of any matrix by hand.