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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. == Decomposition of endomorphisms == 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: :''x'' = ''x''ss + ''x''''n'', 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'', . If ''x'' is an invertible operator, then a multiplicative Jordan–Chevalley decomposition expresses ''x'' as a product: :''x'' = ''x''ss · ''x''u, 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''. For endomorphisms of a finite dimensional vector space whose characteristic polynomial splits into linear factors over the ground field (which always happens if that is an algebraically closed field), the Jordan–Chevalley decomposition exists and has a simple description in terms of the Jordan normal form. If ''x'' is in the Jordan normal form, then ''x''ss is the endomorphism whose matrix on the same basis contains just the diagonal terms of ''x'', and ''x''n is the endomorphism whose matrix on that basis contains just the off-diagonal terms; ''x''u is the endomorphism whose matrix is obtained from the Jordan normal form by dividing all entries of each Jordan block by its diagonal element. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Jordan–Chevalley decomposition」の詳細全文を読む スポンサード リンク
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