---
tag: 0ZND7NQ970
kind: definition
name: Eigenvalue
msc: [15A18]
defines: 0BQAA2BK3Q
uses: [0PTF1PTJK0]
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 $F$, and let $T \colon V \to V$ be an [[endomorphism|0J7H0YBFDJ]].
A scalar $\lambda \in F$ is an *eigenvalue* of $T$ when there is a vector $v \in V$ with $v \ne 0$ and
$$T(v) = \lambda v.$$
An eigenvalue of an $n \times n$ matrix $A$ over $F$ is an eigenvalue of the map $F^n \to F^n$, $x \mapsto Ax$.

## Remarks

So $\lambda$ is an eigenvalue of $T$ if and only if $T - \lambda \id_V$ is not injective.
In finite dimensions, this holds if and only if $\det(T - \lambda \id_V) = 0$.
The eigenvalues of the diagonal matrix $\begin{pmatrix} 2 & 0 \\ 0 & 3 \end{pmatrix}$ are $2$ and $3$.
The rotation of $\R^2$ by $90$ degrees has no real eigenvalue.

## Explanations

### L1 human

Most vectors change direction when a linear map acts on them.
An eigenvalue is a scaling factor $\lambda$ such that some nonzero vector only gets stretched by $\lambda$, without a change of its line.
These special directions often reveal the structure of the map.