Suppose that $F$ is a subfield of a field $E$ and, for

$n\times n$ matrices $A_1,\dots,A_m, B_1,\dots,B_m$
over $F$, there exists a matrix $T\in{\rm GL}_n(E)$
such that $T^{-1}A_iT=B_i$ for all $i$.

Does this imply that such a matrix $T$ can be chosen from ${\rm GL}_n(F)$?

It is easy to see that the answer is

**yes**if $m=1$;- and
**yes**if the field $F$ is infinite.

yes. $\endgroup$ – Anton Klyachko Aug 14 '20 at 11:129more comments