---
title: "Eigenvalues and Eigenvectors"
description: "Some directions survive a map unchanged except for scaling. Finding them means solving the characteristic equation, and they explain what the map does far better than its entries do."
canonical: https://lightmysky.com/learn/mathematics/eigenvalues-and-eigenvectors-mt_TVqqaw11qa
source: https://lightmysky.com/learn/mathematics/eigenvalues-and-eigenvectors-mt_TVqqaw11qa.md
retrieved: 2026-09-12
---

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# Eigenvalues and Eigenvectors

Some directions survive a map unchanged except for scaling. Finding them means solving the characteristic equation, and they explain what the map does far better than its entries do.

Subject: Mathematics · Area: Linear Algebra · Ages 19 to 20
Page: https://lightmysky.com/learn/mathematics/eigenvalues-and-eigenvectors-mt_TVqqaw11qa

## Ready when they can

- Find eigenvalues from the characteristic polynomial and eigenvectors from the null space
- Interpret an eigenvector geometrically for a rotation, reflection or shear
- Give a real matrix with no real eigenvalues and explain the geometry

## Lesson: Directions a matrix only stretches

An eigenvector is a direction the map only stretches: A v = lambda v for some scalar lambda, the eigenvalue. They belong to the map, so they survive a change of basis and show what it does. To find them, solve det(A minus lambda I) = 0 for lambda, then solve (A minus lambda I)v = 0 for the vectors. Diagonal matrices show the answer directly: [[4, 0], [0, 1]] has eigenvalues 4 and 1.

**Example.** Work a 2 by 2 fully. A = [[6, 2], [0, 4]] has characteristic polynomial lambda squared minus 10 lambda + 24 = (lambda minus 6)(lambda minus 4), so the larger eigenvalue is 6. And v = (1, 0) with A = [[2, 1], [0, 3]] gives A v = (2, 0) = 2v, an eigenvector with eigenvalue 2.

Geometry explains the zoo. Reflections fix their mirror line and flip the perpendicular one. Shears fix one line and slide parallel to it, with eigenvalue 1 on the fixed direction. Rotations other than half turns fix no real line: the quarter turn has lambda squared + 1 = 0, no real eigenvalues.

Two habits keep you safe. Scaling an eigenvector keeps it an eigenvector, so report one clean vector per eigenspace. And the zero vector never counts, since it stretches trivially under everything. Each solution space is the eigenspace for its eigenvalue.

**Recap.** Solve the characteristic equation for the stretch factors, then solve one homogeneous system per factor for the directions.

## Practice

14 questions on this page, each with its working shown.

## Needs first

- [The Imaginary Unit and Complex Arithmetic](https://lightmysky.com/learn/mathematics/the-imaginary-unit-and-complex-arithmetic-mt_Faemafn550)
- [Determinants and What They Measure](https://lightmysky.com/learn/mathematics/determinants-and-what-they-measure-mt_fWr2D_1BdM)
- [Change of Basis](https://lightmysky.com/learn/mathematics/change-of-basis-mt_HTGuJhX484)

## Opens up

- [Systems of Differential Equations and Eigenvalue Solutions](https://lightmysky.com/learn/mathematics/systems-of-differential-equations-and-eigenvalue-solutions-mt_BbOqJ4S1ur)
- [Operators, Eigenvalues and Measurement](https://lightmysky.com/learn/science/operators-eigenvalues-and-measurement-mt_E9C_9DlwD2)
- [Stationary Distributions and Long-Run Behaviour](https://lightmysky.com/learn/mathematics/stationary-distributions-and-long-run-behaviour-mt_MGfHvUGkeh)
- [Symmetric Matrices and the Spectral Theorem](https://lightmysky.com/learn/mathematics/symmetric-matrices-and-the-spectral-theorem-mt_sdQv4m7Nbk)
- [Coupled Oscillators and Normal Modes](https://lightmysky.com/learn/science/coupled-oscillators-and-normal-modes-mt_vnTGOIqZzP)
- [Diagonalisation and Powers of a Matrix](https://lightmysky.com/learn/mathematics/diagonalisation-and-powers-of-a-matrix-mt_w15uCVhdhX)