linear algebra – Is simultaneous similarity of matrices independent from the base field?

Suppose that $$F$$ is a subfield of a field $$G$$ and, for
$$ntimes n$$ matrices $$A_1,dots,A_m, B_1,dots,B_m$$
over $$F$$, there exists a matrix $$Tin{rm GL}_n(G)$$
such that $$T^{-1}A_iT=B_i$$ for all $$i$$.

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

Surely,

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