primary decomposition theorem

Factor the minimal polynomial of T into powers of distinct irreducibles: m(x) = p1(x)^e1 * p2(x)^e2 * ... * pk(x)^ek. The primary decomposition theorem says V then splits, as an invariant direct sum, into the kernels of those factors — one invariant summand per prime power.

Precisely, set Wi = ker( pi(T)^ei ). The theorem asserts V = W1 (+) W2 (+) ... (+) Wk, each Wi is T-invariant, and the minimal polynomial of T restricted to Wi is exactly pi(x)^ei. Over the complex numbers each pi(x) is (x - lambda_i), so Wi = ker( (T - lambda_i I)^ei ) is the generalized eigenspace for lambda_i.

This is the first, field-honest decomposition: it works over any field, needs no eigenvalues to actually exist, and reduces the study of T to operators whose minimal polynomial is a single prime power. Those are the nilpotent-like pieces that Jordan and rational form then break down further. The proof is constructive — the projections onto each Wi are polynomials in T, built from the partial-fraction pieces of 1/m(x).