# The relation between rational forms and Jordan forms.

Is there any algorithm / method that allows one to determine the Jordan form of a matrix after determining its rational form?

## 1 Answer

I explain the relationship between Jordan and rational forms in these recently written notes. Here are the main points:

1) Jordan form is only available when a transformation is split, i.e., when its minimal polynomial (equivalently, its characteristic polynomial) is a product of linear factors over the ground field.

2) In this case, the rational canonical form takes the form of a direct sum of companion matrices for polynomials $(t-\lambda)^a$. In the lingo of the notes, we have expressed $T$ as a direct sum of cylic, primary invariant subspaces. On each such subspace (or "square block" of the corresponding matrix) the transformation $T-\lambda 1_V$ is nilpotent, so its rational canonical form is that of the companion matrix of the polynomial $t^a$: it has all zeros except $1$'s along the main diagonal. Adding back $\lambda 1_V$, we get a matrix which has $\lambda$'s on the main diagonal, $1$'s on the subdiagonal and otherwise zeros. This is a Jordan block.

3) Or possibly your convention on Jordan blocks is that the $1$'s should appear on the "superdiagonal" rather than the "subdiagonal". This is an inessential difference: if you just reverse the order of the basis elements you will go from my version of Jordan normal form to yours. I prefer mine I suppose because it is that much more closely related to rational canonical form and thus seems more natural.

Here is another way to say this: the rational normal form is more general: it exists for any transformation. Knowing the rational normal form is equivalent to knowing a decomposition of $V$ into a direct sum of cyclic "primary" subspaces -- the latter means that the minimal polynomial is a prime power. The minimal polynomials of these cyclic primary subspaces are precisely the elementary divisors $p_1,\ldots,p_n$ When you have the elementary divisors, the rational canonical form is a direct sum of companion matrices $C(p_1),\ldots,C(p_n)$. When the minimal polynomial is split, each elementary divisor is of the simple form $(t-\lambda_i)^{a_i}$, and from this data you build the Jordan blocks as well, in a way which can be seen in terms of rational canonical forms as described in 2) above. But fundamentally, both forms are obtained from the elementary divisors and the algorithmic work of finding either form is plainly equivalent to that of finding the elementary divisors.