Rational canonical form

Rational canonical form expresses a linear map as a direct sum of companion matrices of invariant factors, and works over any field.

For a finite-dimensional vector space regarded as an $F[X]$-module through a linear operator, the monic invariant factors $a_1\mid \cdots \mid a_k$ give the decomposition ${⨁}_{i}F[X]/(a_i)$. Their product is the characteristic polynomial and the largest is the minimal polynomial.