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Concept
Jordan–Chevalley decomposition
Summary
In mathematics, the Jordan–Chevalley decomposition, named after Camille Jordan and Claude Chevalley, expresses a linear operator as the sum of its commuting semisimple part and its nilpotent part. The multiplicative decomposition expresses an invertible operator as the product of its commuting semisimple and unipotent parts. The decomposition is easy to describe when the Jordan normal form of the operator is given, but it exists under weaker hypotheses than the existence of a Jordan normal form. Analogues of the Jordan-Chevalley decomposition exist for elements of linear algebraic groups, Lie algebras, and Lie groups, and the decomposition is an important tool in the study of these objects.
Consider linear operators on a finite-dimensional vector space over a field. An operator is semisimple if every T-invariant subspace has a complementary T-invariant subspace (if the underlying field is algebraically closed, this is the same as the requirement that the operator be diagonalizable). An operator x is nilpotent if some power xm of it is the zero operator. An operator x is unipotent if x − 1 is nilpotent.
Now, let x be any operator. A Jordan–Chevalley decomposition of x is an expression of it as a sum
x = xs + xn,
where xs is semisimple, xn is nilpotent, and xs and xn commute. Over a perfect field, such a decomposition exists (cf. #Proof of uniqueness and existence), the decomposition is unique, and the xs and xn are polynomials in x with no constant terms. In particular, for any such decomposition over a perfect field, an operator that commutes with x also commutes with xs and xn.
If x is an invertible operator, then a multiplicative Jordan–Chevalley decomposition expresses x as a product
x = xs · xu,
where xs is semisimple, xu is unipotent, and xs and xu commute. Again, over a perfect field, such a decomposition exists, the decomposition is unique, and xs and xu are polynomials in x. The multiplicative version of the decomposition follows from the additive one since, as is easily seen to be invertible,
and is unipotent.
Official source
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