The Jordan-Chevalley decomposition is a theorem in linear algebra concerning the structure of endomorphisms (or linear transformations) on a finite-dimensional vector space. It provides a way to decompose a linear operator into two simpler components: one that is semisimple and one that is nilpotent.
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Over an algebraically closed field, a linear operator has a unique decomposition into a commuting diagonalisable endomorphism and nilpotent endomorphism . On its generalized eigenspace for , set and . The Chinese remainder theorem makes both parts polynomials in , so they preserve every -invariant subspace. Uniqueness follows by restricting any other commuting decomposition to those generalized eigenspaces. Over a perfect field the semisimple part need only become diagonalizable after scalar extension.