---
tag: 0B5H4208Z9
kind: definition
name: Eigenvector
msc: [15A18]
defines: 0N6960W0BV
uses: [0ZND7NQ970]
sources:
  - url: https://en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors
    locator: General definition
---

## Definition

Let $F$ be a field, let $V$ be a vector space over $F$, let $T \colon V \to V$ be an endomorphism, and let $\lambda \in F$.
An *eigenvector* of $T$ for $\lambda$ is a vector $v \in V$ with $v \ne 0$ and $T(v) = \lambda v$.
Then $\lambda$ is an [[eigenvalue|0BQAA2BK3Q]] of $T$.

## Remarks

Each nonzero multiple of an eigenvector is again an eigenvector for the same eigenvalue.
For $A = \begin{pmatrix} 2 & 1 \\ 0 & 3 \end{pmatrix}$, the vector $(1, 1)^T$ is an eigenvector for $3$, because $A (1, 1)^T = (3, 3)^T$.
The zero vector is never an eigenvector, by convention.

## Explanations

### L1 human

An eigenvector keeps its line under the map: the map only stretches it, shrinks it or flips it.
The stretch factor is the eigenvalue.
Consider a reflection of the plane in a line.
The vectors on the line are eigenvectors for $1$, and the vectors perpendicular to it are eigenvectors for $-1$.