Let V be a finite dimensional vector space over an algebraically closed field F, e.g., the field of complex numbers. Then every linear operator T on V can be written as the sum of a diagonalizable operator D and a nilpotent operator N which commute. These operators D and N are unique and each is a polynomial in T.

From these results, one sees that the study of linear operators on vector spaces over an algebraically closed field is essentially reduced to the study of nilpotent operators. For vector spaces over non algebraically closed fields, we still need to find some substitute for characteristic values and vectors. It is a very interesting fact that these two problems can be handled simultaneously and this is what we shall do in the next chapter.

In concluding this section, we should like to give an example which illustrates some of the ideas of the primary decomposition theorem. We have chosen to give it at the end of the section since it deals with differential equations and thus is not purely linear algebra.

In the primary decomposition theorem, it is not necessary that the vector space V be finite dimensional, nor is it necessary for parts (a) and (b) that p be the minimal polynomial for T. If T is a linear operator on an arbitrary vector space and if there is a monic polynomial p such that ** f, then parts (a) and (b) of Theorem 12 are valid for T with the proof which we gave.