Rational Decomposition Theorem?

So I am working through Hoffman and Kunze, and in the chapter on canonical forms I am having some difficulty due to the lack of examples. Using another book, I figured out how to calculate the Rational and Jordan canonical forms for a given matrix (linear operator). And I assume that the so called "rational decomposition" has something to do with the Rational Canonical Form? But what exactly is the relationship?

For a finite dimensional vector space V and linear operator T, There exists r non-zero vectors in V and r respective T-Annihilators so that the direct sum of the cyclic subspace generated by the vectors with respect to T equals V AND the T Annihilator for each vector divides the next one.

Hard to say in words, but I couldn't find the LaTex stuff. I haven't been on here in a while, did that go away?

Anyways, like I said, I am sure it has something to do with the canonical form, just hoping someone can point me in the right direction. I am currently trying to figure out from the proof of the rational decomposition theorem where he is getting the vectors, but haven't yet seen the connection to the canonical form.

Any help would be greatly appreciated.