primary rational canonical form

The primary rational canonical form is the second flavor of RCF — the one organized around elementary divisors instead of invariant factors. Instead of a single companion block per invariant factor, you use one companion block per prime-power factor p^e. The result is still block-diagonal of companion matrices, just chopped into finer, prime-localized pieces.

It comes from the elementary-divisor version of the structure theorem: V decomposes as a direct sum of cyclic modules F[x]/(p_i^(e_i)), one per elementary divisor, and each summand contributes the companion block C(p^e). When the prime p is linear, p = x - lambda, this companion block is similar to a Jordan block — so over an algebraically closed field the primary RCF is just a step away from JCF.

Why keep both forms? The ordinary (invariant-factor) RCF has the fewest blocks and is the cleanest canonical representative; the primary form exposes the prime structure, separating the contributions of different irreducible factors so you can analyze each prime independently. It is the bridge object: literally the same decomposition the Jordan form uses, but written with companion blocks so it survives over any field.

A caveat on canonicity. Both forms are canonical, but they are canonical for different bookkeeping. The blocks of the primary form can be permuted (any ordering of the elementary divisors), so to make it a strict normal form you fix a convention — group by prime, then by increasing exponent. The underlying multiset of elementary divisors is what is truly invariant.