 Rational canonical form

The rational canonical form of a linear operator on a finite-dimensional vector space over a field is a direct sum $C(f_1)\oplus \cdots \oplus C(f_r)$ of companion matrices, where the monic nonconstant invariant factors of a linear operator satisfy $f_1\mid \cdots \mid f_r$. Their degrees sum to the dimension of the space. It exists over any field and classifies similar matrices, without requiring their eigenvalues to lie in that field. It comes from the structure theorem for finitely generated modules over a principal ideal domain, applied to the module in which the indeterminate acts as the given operator.