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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?

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    $\begingroup$ For each block of the rational canonical form, i.e. the companion matrix associated to the polynomial $(t-\lambda)^m$, you just need to replace the last column with zeros and replace the diagonal with $\lambda$s. $\endgroup$ – Daniel May 2 '15 at 23:34

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