If $A$ is a matrix over a field whose characteristic polynomial splits, then how is the Jordan form related to the rational Canonical form and can we recover one from the other in a computationally mechanical way?
I expect that each Jordan Block corresponding to the eigenvalue $\lambda$ is similar to each Companion matrix of the irreducible polynomials $(t-\lambda)^m$, but how can I go from the Rational Canonical Form to the Jordan Form simply?