The unique expression of a linear endomorphism of a finite-dimensional vector space (over a field where the minimal polynomial splits and is separable) as the sum of a semisimple (diagonalizable) endomorphism and a nilpotent endomorphism that commute with each other.

Principle

Factor the minimal polynomial into relatively prime factors for the semisimple and nilpotent parts; construct the semisimple part as a polynomial in the endomorphism projecting to the diagonalizable factor and the nilpotent part as the remainder, ensuring commutation and uniqueness.

Demonstration

For a matrix A over an algebraically closed field, conjugate A to its Jordan normal form J = D + N where D is block-diagonal with eigenvalues on the diagonal (semisimple) and N is strictly block-upper-triangular (nilpotent). Then A = S + N with S similar to D and [S,N]=0.

Misapplication

Assuming the decomposition exists over an arbitrary base field without checking that the minimal polynomial splits into separable factors; another misuse is treating the Jordan–Chevalley decomposition as a choice of basis-dependent Jordan form rather than an intrinsic split into commuting parts.

Consequence

Gives an intrinsic separation of an operator into diagonalizable and nilpotent behavior, facilitating computations of functions of operators (exponential, polynomials) and simplification in representation theory and algebraic group actions.

Reversal

The inversion would be to conflate semisimple and nilpotent behavior by taking the radical or semisimple quotient instead of splitting additively; this loses the commuting structure the decomposition preserves.

Boundary

Requires the minimal polynomial to split and be separable or a perfect base field (e.g., characteristic zero or finite fields). Over imperfect fields or without splitting, one may need to pass to a field extension or work with a multiplicative version (Jordan decomposition in algebraic groups).

Semantic Tension

Often confused with Jordan canonical form: the latter depends on a chosen basis and lists block structure, whereas Jordan–Chevalley is an intrinsic operator-level splitting into commuting semisimple and nilpotent parts independent of basis when hypotheses hold.

Synthesis

The Jordan–Chevalley decomposition is the canonical, basis-independent splitting of an endomorphism into commuting semisimple and nilpotent summands obtained from the factorization of the minimal polynomial, isolating diagonalizable action from nilpotent behavior.