We are trying to study a linear operator T on the finite dimensional space V, by decomposing T into a direct sum of operators which are in some sense elementary. We can do this through the characteristic values and vectors of T in certain special cases, i.e., when the minimal polynomial for T factors over the scalar field F into a product of distinct monic polynomials of degree 1. What can we do with the general T? If we try to study T using characteristic values, we are confronted with two problems. First, T may not have a single characteristic value; this is really a deficiency in the scalar field, namely, that it is not algebraically closed. Second, even if the characteristic polynomial factors completely over F into a product of polynomials of degree 1, there may not be enough characteristic vectors for T to span the space V; this is clearly a deficiency in T. The second situation is illustrated by the operator T on ** f (F any field) represented in the standard basis by ** f. The characteristic polynomial for A is ** f and this is plainly also the minimal polynomial for A (or for T). Thus T is not diagonalizable. One sees that this happens because the null space of ** f has dimension 1 only. On the other hand, the null space of ** f and the null space of ** f together span V, the former being the subspace spanned by ** f and the latter the subspace spanned by ** f and ** f.