existence and uniqueness of RCF

This is the theorem that makes the rational canonical form worth caring about. It says two things at once: existence — every operator on a finite-dimensional space over any field F is similar to a matrix in rational canonical form; and uniqueness — that RCF is the only one (the invariant factors, hence the companion-block layout, are completely determined by the operator).

Existence comes from applying the structure theorem to V as an F[x]-module: V splits as a direct sum of cyclic modules F[x]/(f1) plus ... plus F[x]/(fk) with f1 | f2 | ... | fk, and choosing the natural basis in each cyclic piece turns T into the block-diagonal of companion matrices. Uniqueness comes from the uniqueness of the invariant factors, which are the Smith normal form diagonal of xI - A — gcds of minors, hence basis-independent.

Combine the two halves and you get a complete invariant for similarity: A and B are similar over F if and only if they have identical invariant factors, equivalently identical RCF. No eigenvalues, no diagonalizability assumptions, no algebraically closed field required — it works uniformly over Q, R, finite fields, anything.

One precise caveat about uniqueness. The invariant-factor list (and therefore the RCF) is unique, but the actual cyclic subspaces realizing the decomposition are not unique — only their isomorphism types are. So uniqueness is a statement about the canonical matrix, not about a canonical choice of decomposing subspaces.