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For endomorphisms of a finite dimensional vector space over an algebraically closed field, there is a simple description in terms of the [[Jordan normal form]]. If ''x'' is in the Jordan normal form, then ''x''<sub>ss</sub> is the matrix containing just the diagonal terms of ''x'', ''x''<sub>n</sub> is the matrix containing just the off-diagonal terms, and ''x''<sub>u</sub> is the matrix with the diagonal entries replaced by 1s. |
For endomorphisms of a finite dimensional vector space over an algebraically closed field, there is a simple description in terms of the [[Jordan normal form]]. If ''x'' is in the Jordan normal form, then ''x''<sub>ss</sub> is the matrix containing just the diagonal terms of ''x'', ''x''<sub>n</sub> is the matrix containing just the off-diagonal terms, and ''x''<sub>u</sub> is the matrix with the diagonal entries replaced by 1s. |
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==Connection to [[Jordan normal form]]== |
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Jordan–Chevalley decomposition is an abstract form of the Jordan normal form of a matrix. Jordan normal form ''J'' of a matrix ''A'' satisfies ''A'' = ''MJM''<sup>−1</sup> for some invertible matrix ''M'' and ''J'' = ''D'' + ''N'', where ''D'' is a diagonal matrix and ''N'' is a matrix with the only nonzero terms in the first codiagonal, thus ''N'' is nilpotent. Then ''A'' = ''MDM''<sup>−1</sup> + ''MNM''<sup>−1</sup> is a decomposition of ''A'' into a semisimple matrix and nilpotent matrix. |
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== References == |
== References == |
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Revision as of 16:47, 11 January 2009
In mathematics, the Jordan–Chevalley decomposition, named after Camille Jordan and Claude Chevalley, also known as Dunford decomposition, named after Nelson Dunford, 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.
Consider linear operators on a finite-dimensional vector space. An operator is semisimple if the roots of its minimal polynomial are all distinct (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. The Jordan–Chevalley decomposition expresses x as a sum:
- x = xss + xn,
where xss is semisimple, xn is nilpotent, and xss and xn are polynomials in x (in particular they commute). This decomposition is unique, in that if x = s + n with s semisimple, n nilpotent, and sn = ns, then s = xss and n = xn.[1]
If x is an invertible operator, then the Jordan–Chevalley decomposition expresses x as a product:
- x = xss · xu,
where xss is semisimple, xu is unipotent, and xss and xu are polynomials in x (in particular they commute). This decomposition is unique, in that if x = s · u with s semisimple, u nilpotent, and su = us, then s = xss and u = xu.
For endomorphisms of a finite dimensional vector space over an algebraically closed field, there is a simple description in terms of the Jordan normal form. If x is in the Jordan normal form, then xss is the matrix containing just the diagonal terms of x, xn is the matrix containing just the off-diagonal terms, and xu is the matrix with the diagonal entries replaced by 1s.
References
- Humphreys, James E. (1972), Introduction to Lie Algebras and Representation Theory, Springer, ISBN 978-0-387-90053-7.
- Serge Lang, Algebra (3 ed), Addison-Wesley, 1993, ISBN 0-201-55540-9. Chap.XIV.2, p.559.
- Dunford, Nelson, Direct decompositions of Banach spaces. Bol. Soc. Mat. Mexicana 3, (1946), 1-12. MR21240
Notes
- ^ Humphreys, proposition 4.2, p. 17