---
tag: 0PTF1PTJK0
kind: definition
name: Endomorphism
msc: [15A04]
defines: 0J7H0YBFDJ
uses: [0RJF668P1T]
sources:
  - url: https://en.wikipedia.org/wiki/Endomorphism
    locator: lead section
---

## Definition

Let $F$ be a field and let $V$ be a vector space over $F$.
An *endomorphism* of $V$ is a [[linear map|0NRYHJPQG2]] $T \colon V \to V$.
The set of all endomorphisms of $V$ is written $\End_F(V)$.

## Remarks

An endomorphism is also called a *linear operator* on $V$.
Endomorphisms can be composed with each other, so their powers $T^k$ and polynomials $p(T)$ are defined.
Under composition and addition, $\End_F(V)$ is the [[algebra of endomorphisms|0E3X2G1ZQB]].
Eigenvalues, the trace and the determinant are defined for endomorphisms of finite-dimensional spaces.

## Explanations

### L1 human

An endomorphism maps a space into itself.
So we can apply it again and again, and compare a vector with its image.
In a basis, an endomorphism of an $n$-dimensional space is a square $n \times n$ matrix.