Characterization of invariant subspaces for a nilpotent linear operator that admit complementary invariant subspaces

The aim of this work is to solve the problem of determining the necessary and sufficient conditions for a vector subspace invariant by a nilpotent endomorphism to admit a complementary invariant subspace for the same linear operator. As applications, we offer results about Jordan bases associated with nilpotent linear maps and reflexive generalized inverses of finite potent endomorphisms and square matrices.