Linear and multilinear algebra; matrix theory Canonical forms and decompositions Theorem

Jordan–Chevalley decomposition

Builds on Semisimple endomorphism Nilpotent endomorphism

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:

$T = S + N$ .
$S$ is semisimple and $N$ is nilpotent.
$SN = NS$ .

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 $N = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}$ .

Reading path

To understand this page from the ground up, read these 8 pages in order. The foundations come first.

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Vector space definition
Linear map definition
Linear subspace definition
Endomorphism definition
Nilpotent endomorphism definition
Invariant subspace definition
Semisimple endomorphism definition
Jordan–Chevalley decomposition statement

Tag	`011K11VPJN`
URI	`m2://011K11VPJN`
Kind	statement, theorem
Writing	W0 stub
MSC	15A21, 15A20
Status	This copy of the site has no build data. Run `m2 status 011K11VPJN`.
Uses	Semisimple endomorphism, Nilpotent endomorphism
Same as	Wikidata Q1631086 Wikipedia Jordan–Chevalley decomposition

Jordan–Chevalley decomposition

Statement

Remarks

Reading path

Further reading

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