Eigenvectors in action

Apply a matrix to most vectors and they swing off in new directions. A special few only stretch — same direction, scaled length. These eigenvectors are the transformation's grain, and finding them is how you predict long-run behavior of anything linear.

Build the intuition

Stubborn directions

An eigenvector v of matrix A satisfies Av = λv: the action only rescales it by the eigenvalue λ. Stretch ×2 along one axis: that axis is an eigenvector with λ = 2. A reflection's mirror line: eigenvector with λ = 1; the perpendicular, λ = −1. The eigenvectors are the directions the transformation respects.

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

Repeat the map and eigen wins

Apply A over and over — a population year by year, a webpage random-surfer step by step — and whichever eigenvector has the largest |λ| gradually dominates: components along it grow fastest (λ > 1), or shrink slowest (λ < 1). Long-run behavior is an eigenvalue contest. λ's magnitude also answers stability: all |λ| < 1 means repeated application dies out — the unit-circle rule you'll meet for poles in discrete systems.

Diagonalize: the axes where life is easy

Describe a transformation in its eigenvector basis and the matrix becomes diagonal — each coordinate evolving independently, just multiplied by its λ. Coupled mess becomes parallel simplicity. Vibration analysis (modes), quantum mechanics (states), and Fourier analysis (LTI systems diagonalized by sinusoids) are all the same move: switch to the eigen-basis, watch the problem fall apart.

See it move

InteractiveA matrix is a verb

Rotate 45°: The whole plane turns together. Every game camera move is a cousin of this matrix. Matrix: [0.71, -0.71; 0.71, 0.71]

Watch “Squash flat”: the horizontal axis survives untouched (λ = 1) while vertical shrinks (λ = 0.35). You're seeing eigenvectors — the directions the grid doesn't rotate.

A worked example

A two-city migration

Each year, 10% of city A's residents move to B and 20% of B's move to A. Will populations churn forever?
The yearly map has eigenvalues λ = 1 and λ = 0.7. The λ = 1 eigenvector is the (2 : 1) split — apply the map and that split returns unchanged.
Every starting condition's λ = 0.7 component shrinks 30% yearly, so populations settle to A:B = 2:1 — equilibrium found without simulating a single year. The eigenvector was the answer key.

Out in the world

PageRank: the web's eigenvector

Model a random surfer clicking links forever; the long-run fraction of time on each page is the dominant eigenvector of the web's link matrix. Google's founding insight was that this eigenvector is a ranking. A trillion-dollar company's seed was Av = λv.

Common confusion, cleared

“Eigenvalues are abstract bookkeeping with no physical meaning.”

They're growth rates, decay rates, vibration frequencies, and stability verdicts. When engineers say a bridge mode “rings at 2.3 Hz,” that's an eigenvalue talking.

“Every matrix conveniently has real, friendly eigenvectors.”

Rotations, for instance, have complex eigenvalues — their “stretch-only directions” live in complex space, encoding the rotation as a phase. (Which is exactly why complex exponentials diagonalize oscillating systems.)

Check yourself

PracticeQuick check

A system's update matrix has eigenvalues 0.9 and 0.4. Repeated updates…

Recap

Eigenvectors only rescale under the map; eigenvalues are the factors.
Repeated application is ruled by the largest |λ| — growth, decay, stability.
In the eigen-basis everything decouples; Fourier is this idea applied to LTI systems.

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