---
tag: 0EP59ESQ94
kind: definition
name: Invariant subspace
msc: [15A04, 47A15]
defines: 0ACX5QT8TE
uses: [0PTF1PTJK0, 0EPTJA5QV4]
sources:
  - url: https://en.wikipedia.org/wiki/Invariant_subspace
    locator: For a single operator
---

## 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 [[subspace|00YC1D652K]] $W$ of $V$ is *invariant* under $T$, or *$T$-invariant*, when $T(w) \in W$ for all $w \in W$.

## Remarks

The subspaces $0$ and $V$, $\ker T$ and $\im T$, and each [[eigenspace|0G0RFPG8HR]] of $T$ are $T$-invariant.
A line spanned by $v \ne 0$ is $T$-invariant if and only if $v$ is an eigenvector of $T$.
If $V = W_1 \oplus W_2$ with both $W_i$ invariant, then the matrix of $T$ in an adapted basis is block diagonal.

## Explanations

### L1 human

An invariant subspace is a part of the space that the map never leaves.
Vectors inside it stay inside it.
We can then study the map on this smaller space alone.