# Eigenvalue & Eigenvector

**Type:** Concept  
**Domain:** Algebra  
**Codex URL:** /mathematics/algebra/eigenvalue-and-eigenvector/  
**Entry status:** Live — v1.0 (2026-05-27)

## Summary
Eigenvectors are directions a matrix only scales, never rotates; eigenvalues are the scaling factors.

## Defining equation
Av = λv

## Finding eigenvalues
det(A−λI)=0 — the characteristic polynomial's roots are the eigenvalues.

## Diagonalisation
A=PDP⁻¹ when n independent eigenvectors exist — makes computing Aⁿ trivial.

## Spectral theorem
Real symmetric matrices: always real eigenvalues, always orthogonally diagonalisable.

## PageRank
Dominant eigenvector (eigenvalue=1) of the web's link matrix = steady-state importance ranking (Page, Brin, Motwani, Winograd, 1999).

## Quantum mechanics
Observables = Hermitian operators; measured values = eigenvalues; post-measurement state = eigenvector.

## Perron-Frobenius theorem
Non-negative matrices have a uniquely largest positive eigenvalue with non-negative eigenvector (Perron 1907, Frobenius 1912).

## Sources
### Tier 1
- Strang, G. (2016). *Introduction to Linear Algebra*. 5th ed.
- Horn, R.A. and Johnson, C.R. (2012). *Matrix Analysis*. 2nd ed.
### Tier 2
- Page, L. et al. (1999). The PageRank Citation Ranking. Stanford InfoLab.

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*Mathematics Codex entry v1.0 — added 2026-05-27 — thecodex.expert/mathematics/*