rational canonical form

Jordan form is wonderful but it cheats: it needs the eigenvalues to actually exist, which forces you to work over the complex numbers (or some field where the characteristic polynomial fully factors). The rational canonical form is the honest alternative. It produces a standard, block-diagonal matrix for any operator over ANY field — rationals, reals, finite fields — without ever computing a single eigenvalue or leaving the field you started in.

Concretely, every operator T on a finite-dimensional space V is similar to a unique block-diagonal matrix whose blocks are companion matrices C(f1), C(f2), ..., C(fk) of polynomials f1 | f2 | ... | fk (each dividing the next). Those polynomials are the invariant factors of T, and the companion blocks are stacked on the diagonal. The word rational means it is built using only the four field operations on the entries of the original matrix — no roots, no extension.

It matters because it is a genuine canonical form for similarity: two matrices over a field F are similar over F if and only if they have the exact same rational canonical form. So RCF settles the similarity problem completely, and it does so field-independently — the answer does not change when you climb up to a bigger field.

A caveat about the name: rational refers to staying inside the base field, not to rational numbers specifically. Over the complex numbers RCF still exists; it is just usually less convenient than Jordan form there, because companion blocks are bulkier than Jordan blocks when eigenvalues are available.