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.