Canonical form of pairs of matrices over arbitrary field

I'm looking for a canonical form of pairs of matrices over arbitrary field up to equivalence (Calling pairs ($A, B$) and ($A_1, B_1$) over a field F equivalent if invertible C and D exist over F such that $AC = DA_1$ and $BC = DB_1$). Unfortunately, I was only able to find the answer in the case if F is algebraically closed fields of characteristic 0.

If such a general theory exists (regardless of the field characteristic or whether it is algebraically closed or not), could you please point me to a source?