Dunford decomposition

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 parts. The multiplicative decomposition expresses an invertible operator as the product of its commuting semisimple and unipotent parts. The decomposition is important in the study of algebraic groups. 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.

Consider linear operators on a finite-dimensional vector space over a perfect field. An operator T 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 x^m 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:

where x_ss is semisimple, x_n is nilpotent, and x_ss and x_n commute. If such a decomposition exists it is unique, and x_ss and x_n are in fact expressible as polynomials in x, (Humphreys 1972, Prop. 4.2, p. 17).

If x is an invertible operator, then a multiplicative Jordan–Chevalley decomposition expresses x as a product:

where x_ss is semisimple, x_u is unipotent, and x_ss and x_u commute. Again, if such a decomposition exists it is unique, and x_ss and x_u are expressible as polynomials in x.

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