Fitting decomposition

Take any operator T and iterate it. The images shrink, ker(T) inside ker(T^2) inside ... grows, im(T) contains im(T^2) contains ... shrinks. In finite dimensions both chains must stabilize: past some power n both stop changing. Fitting's idea is to split V using exactly these eventual, stabilized pieces.

Let n = dim V. Set K = ker(T^n), the eventual kernel, and I = im(T^n), the eventual image. Fitting's lemma says V = K (+) I, both are T-invariant, T restricted to K is nilpotent (it eventually kills everything in K), and T restricted to I is invertible. So every operator is, on the nose, a nilpotent part plus an invertible part on complementary invariant subspaces.

This is the cleanest, most field-agnostic way to peel off nilpotency — no eigenvalues, no factoring of polynomials, just stabilizing kernels and images. Applied to T - lambda I instead of T, the eventual kernel becomes the generalized eigenspace for lambda, so Fitting at each eigenvalue is the engine of primary decomposition. It also drives the Jordan-Chevalley split: K carries the part where the semisimple piece is scalar lambda and N is nontrivial.