Jordan-Chevalley decomposition

Jordan form puts an operator into nearly-diagonal blocks with ones on a superdiagonal. The Jordan-Chevalley decomposition is the coordinate-free distillation of that: it splits T cleanly into a diagonalizable-in-spirit part plus a nilpotent part, T = S + N, with the two parts cooperating rather than interfering.

Precisely, over a perfect field (e.g. C or R or any field of characteristic zero) there exist a semisimple operator S and a nilpotent operator N with T = S + N and SN = NS. The commuting condition is essential — it is what makes the split meaningful and unique. Both S and N are polynomials in T with no constant term needed beyond what T supplies, so they automatically preserve every T-invariant subspace.

Uniqueness is the headline: there is exactly one such pair. You build it from primary decomposition — on each generalized eigenspace S acts as the scalar lambda and N = T - lambda I is nilpotent. The payoff is computational: exp(T) = exp(S) exp(N) because S and N commute, and exp(N) is a finite polynomial since N is nilpotent. This is the standard route to solving linear ODE systems and underlies algebraic-group theory.