I have been studying Cooperstein's Advanced Linear Algebra for about seven months now and I am having problems understanding how to find the elementary divisors of a linear operator and how to find the invariant factors of a linear operator as well. I feel as though if I'm having trouble with these then I won't be able to move forward. I understand that the invariant factors comprise the characteristic polynomial and can find them using determinants and eigenvalues (I get that), but the text builds the theory up using direct sums, minimal polynomials of T, and T-cyclic subgroups and I'm just not seeing it.
As for an example, Let $T$ be an element of $L(\mathbf R^4,\mathbf R^4)$ be the operator given by $$ T(v) = \begin{pmatrix} -3&2&2&-4\\-3&1&4&-4\\-2&0&3&-2\\-1&0&2&-1\end{pmatrix}(v) $$ Determine the elementary divisors and the invariant factors of $T$.
(how do I write the correct matrix in LaTeX?)
Using the Smith normal form algorithm on $T - x I$ you find that the invariant factors (at least, as I am used to call them) are $$ 1,1,1,x^4 -1. $$ (In particular minimal polynomial = characteristic polynomial = $x^4 - 1$.) It follows that over the rationals the elementary divisors are $$ x-1, x+1, x^2 + 1. $$
PS A note on terminology. Wikipedia has the same definitions of invariant factors and elementary divisors I am used to. However, in the article above, the invariant factors are computed, but they are called elementary divisors.