In usual linear algebra (over a field or ring), matrices have several invariants (some more 'numerical' than others) under change of basis - e.g. trace, determinant, eigenvalues - many of these encompassed in the characteristic polynomial.
Consider now a semilinear map over a ring $R$ with a ring automorphism $\sigma:R\to R$. (feel free to assume $\sigma$ is the Frobenius and $R$ is a perfect ring).
Given an $n\times n$ matrix $A$, change of basis by an element $P$ of $\mathbf{GL}_n(R)$ acts as $PA(\sigma(P))^{-1}$, where $\sigma(P)$ is obtained by applying $\sigma$ to every entry of $P$; i.e. by $\sigma$-twisted conjugation.
What 'things' remain invariant? It doesn't seem to make sense to talk about the characteristic polynomial or eigenvalues, but certainly some things do stay invariant such as kernel, rank (whenever defined). I think one can talk about eigenvectors but not eigenvalues!
Thank you.