The Cyclic Decomposition Theorem

Let V be a finite-dimensional F -vector space, and let T : V → V be a linear transformation. In this note we prove that V is a direct sum of cyclic T -invariant subspaces. More specifically, we prove that V is a direct sum of cyclic T -invariant subspaces whose annihilators are generated by powers of irreducible polynomials, and that the collection of these polynomials is uniquely determined. Recall that by using T we can make V into an F [x]-module by defining scalar multiplication by f(x) · v = f(T )(v) for all f(x) ∈ F [x]. We also recall that a T -invariant subspace of V is nothing more than an F [x]-submodule of V . We use module language in this note.