---
tag: 0QS8P74F2Z
kind: definition
name: Semisimple endomorphism
msc: [15A04, 15A20]
defines: 08VRX3Q4PB
uses: [0EP59ESQ94]
sources:
  - url: https://en.wikipedia.org/wiki/Semisimple_operator
    locator: lead section
---

## Definition

Let $F$ be a field, let $V$ be a finite-dimensional vector space over $F$, and let $T \colon V \to V$ be an endomorphism.
The map $T$ is *semisimple* when, for each $T$-[[invariant subspace|0ACX5QT8TE]] $W$ of $V$, there is a $T$-invariant subspace $W'$ with $V = W \oplus W'$.

## Remarks

The map $T$ is semisimple if and only if its minimal polynomial has no repeated irreducible factor.
Over an algebraically closed field, $T$ is semisimple if and only if it is [[diagonalisable|07SQCNV233]].
The rotation of $\R^2$ by $90$ degrees is semisimple, but it is not diagonalisable over $\R$.

## Explanations

### L1 human

Semisimple maps generalise diagonalisable maps to fields where eigenvalues can be missing.
They have no hidden nilpotent part.
Over $\C$, semisimple simply means diagonalisable.