Jordan–Chevalley decomposition

The unique expression of a linear operator as a sum of commuting semisimple and nilpotent operators. Over a suitable field, both components are polynomials in the operator.

Semisimple operator: A linear operator that is diagonalizable over an algebraic closure of its field. This component captures the operator’s eigenvalue action without its nilpotent part.
Linear operator: A linear map from a vector space to itself. The decomposition applies to operators on finite-dimensional vector spaces.
Camille Jordan: A French mathematician known for foundational work in group theory and canonical forms of matrices. Jordan’s canonical-form theory makes the semisimple and nilpotent parts visible block by block.
Jordan–Chevalley decomposition of Lie algebras: The decomposition of an element of a Lie algebra into commuting semisimple and nilpotent parts. It extends the operator split to elements of Lie algebras and their adjoint actions.
Diagonalizable operator: A linear operator with a basis of eigenvectors over the field under consideration. It is purely semisimple, whereas a general operator may also have a nonzero nilpotent component.
Nilpotent operator: A linear operator whose some positive integer power is zero. This component captures the non-diagonalizable behavior while commuting with the semisimple part.
Eigenvalue: A scalar for which a linear operator has a nonzero vector that it scales by that scalar. The semisimple component preserves the operator’s eigenvalues.
Marie Ennemond Camille Jordan: The full name of French mathematician Camille Jordan. This is an alternate name for the same mathematician, not a separate origin of the decomposition.
Jordan–Chevalley decomposition of algebraic groups: The decomposition of an algebraic-group element into commuting semisimple and unipotent parts. It replaces the additive nilpotent component with a multiplicative unipotent component.
Additive Jordan decomposition: The expression of an operator as a sum of commuting semisimple and nilpotent parts. This name emphasizes the additive form, distinguishing it from multiplicative group decompositions.

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Fitting lemmaRelated: Fitting's splitting helps isolate nilpotent behavior in finite-dimensional linear algebra.