Rational Canonical Form

A canonical matrix form over a field that expresses a linear operator as a block diagonal matrix of companion matrices determined by its invariant factors; it classifies similarity classes over the base field without requiring algebraic closure.

A canonical representative for the similarity class of a linear operator over a field, constructed from the invariant factors (monic divisors of the characteristic polynomial) as a direct sum of companion blocks; it classifies matrices up to similarity without assuming polynomial splitting.