---
tag: 011K11VPJN
kind: statement
role: theorem
name: Jordan–Chevalley decomposition
msc: [15A21, 15A20]
uses: [0QS8P74F2Z, 0M8MHYERTY]
sources:
  - url: https://en.wikipedia.org/wiki/Jordan–Chevalley_decomposition
    locator: Decomposition of a linear operator
same_as:
  wikidata: Q1631086
  wikipedia: Jordan–Chevalley decomposition
---

## Statement

Let $F$ be a perfect field, for example a field of characteristic $0$ or an algebraically closed field.
Let $V$ be a finite-dimensional vector space over $F$, and let $T \colon V \to V$ be a linear map.
Then there are unique linear maps $S, N \colon V \to V$ with these properties:

1. $T = S + N$.
2. $S$ is [[semisimple|08VRX3Q4PB]] and $N$ is [[nilpotent|063D5Z2C70]].
3. $SN = NS$.

## Remarks

Moreover, $S$ and $N$ are polynomials in $T$ without constant term.
Over $\C$, $S$ is diagonalisable; in a Jordan basis, $S$ is the diagonal part of the Jordan form.
For $J_2(\lambda)$, the decomposition is $S = \lambda I_2$ and $N = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}$.

## Explanations

### L1 human

Every linear map over $\C$ splits into a diagonalisable part and a nilpotent part.
The two parts commute with each other.
This splitting is unique, and it makes functions of matrices easier to compute.