similarity is field-independent
Here is a striking and useful theorem: if two matrices with entries in a field F become similar after you enlarge F to a bigger field K, then they were already similar over F to begin with. Climbing to a bigger field can never create a new similarity — whatever you discover up high was already true down at the base.
Stated precisely: let A and B have entries in F, and suppose there is an invertible P with entries in K (some extension of F) such that P^-1 A P = B. The conclusion is that there exists such a P with entries in F. The change-of-basis matrix can be brought back down to the base field.
The proof is a one-line consequence of rational canonical form. The invariant factors are computed by the Smith normal form of xI - A, using only gcds of polynomials over F — operations that do not change when you pass to K. So A and B have the same invariant factors over F if and only if they have them over K. Same invariant factors means same RCF means similar over F.
Why it is reassuring: it means the similarity classification you do over your home field is the final answer; you never have to worry that some exotic extension secretly merges two classes. Contrast this with diagonalizability, which genuinely is field-dependent — a matrix can be non-diagonalizable over R yet diagonalizable over C. Similarity itself, though, descends.
Similar over C forces similar over Q: the witnessing change-of-basis can always be found inside the base field.
Contrast with eigenvalues, eigenvectors, and elementary divisors, all of which can change under field extension. Invariant factors — and hence the similarity class — are the robust, field-stable data.