If ** f are the projections associated with the primary decomposition of T, then each ** f is a polynomial in T, and accordingly if a linear operator U commutes with T then U commutes with each of the ** f i.e., each subspace ** f is invariant under U.

In the notation of the proof of Theorem 12, let us take a look at the special case in which the minimal polynomial for T is a product of first degree polynomials, i.e., the case in which each ** f is of the form ** f. Now the range of ** f is the null space ** f of ** f. Let us put ** f. By Theorem 10, D is a diagonalizable operator which we shall call the diagonalizable part of T. Let us look at the operator ** f. Now ** f ** f so ** f. The reader should be familiar enough with projections by now so that he sees that ** f and in general that ** f. When ** f for each i, we shall have ** f, because the operator ** f will then be 0 on the range of ** f.

Let N be a linear operator on the vector space V. We say that N is nilpotent if there is some positive integer r such that ** f.

Let T be a linear operator on the finite dimensional vector space V over the field F. Suppose that the minimal polynomial for T decomposes over F into a product of linear polynomials. Then there is a diagonalizable operator D on V and a nilpotent operator N on V such that (a) ** f, (b) ** f. The diagonalizable operator D and the nilpotent operator N are uniquely determined by (a) and (b) and each of them is a polynomial in T.