--- 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: 2026-11-12 --- > **Agent view.** This is the Markdown twin of the page, for tools and assistants. > When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md > API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md > Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt > Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json > Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`. # 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 scales An eigenvector is a nonzero vector v with A v = lambda v. The eigenvalue lambda can stretch, shrink, reverse or send v to zero. I is the identity matrix. Nonzero solutions of (A minus lambda I)v = 0 exist exactly when det(A minus lambda I) = 0. For A = [[a, b], [c, d]], this determinant is (a minus lambda)(d minus lambda) minus bc. A diagonal matrix has its diagonal entries as eigenvalues: [[4, 0], [0, 1]] has 4 and 1. **Example.** A = [[6, 2], [0, 4]] has characteristic polynomial lambda squared minus 10 lambda + 24 = (lambda minus 6)(lambda minus 4), so its eigenvalues are 6 and 4. For lambda = 4, A minus 4I = [[2, 2], [0, 0]]. Solve 2x + 2y = 0: v = (1, minus 1) gives A v = (4, minus 4) = 4v. For A = [[2, 1], [0, 3]], v = (1, 0) gives A v = (2, 0) = 2v. Diagonal matrices have their diagonal entries as eigenvalues. *(drawing: Solve (A minus 4I)v = 0; any nonzero solution is an eigenvector for 4.)* In the plane, reflection in a line through the origin fixes that line (lambda = 1) and reverses its perpendicular direction (lambda = -1). A nonzero shear fixes one line and slides parallel to it, with lambda = 1. Identity rotations and half turns preserve every line through the origin, with eigenvalues 1 and -1 respectively. Other planar rotations about the origin preserve no real line. A quarter turn has characteristic equation lambda squared + 1 = 0, with no real roots. An eigenspace is the full solution space of (A minus lambda I)v = 0, including zero. Eigenvectors are its nonzero vectors. Nonzero multiples of an eigenvector have the same eigenvalue; multiplying by zero does not give an eigenvector. One vector describes a one-dimensional eigenspace. A higher-dimensional eigenspace needs a basis. For A = 2I, the eigenspace for 2 is the whole plane. Eigenvalues stay the same under a basis change; eigenvector coordinates change. **Recap.** Solve the characteristic equation for the scale factors, then solve one homogeneous system per factor for the directions. *Lesson and practice questions originally authored for this dataset.* ## Practice 14 questions on this page, each with its working shown. ## Needs first - [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) ## Optional connections - [The Imaginary Unit and Complex Arithmetic](https://lightmysky.com/learn/mathematics/the-imaginary-unit-and-complex-arithmetic-mt_Faemafn550) ## 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) - [Diagonalisation and Powers of a Matrix](https://lightmysky.com/learn/mathematics/diagonalisation-and-powers-of-a-matrix-mt_w15uCVhdhX)