Assume $K/F$ be a field extension and $A,B \in M_n(F)$ . If $A,B$ be conjugate in $GL_n(K)$, is it true that $A,B$ are conjugate in $GL_n(F)$?

Question

Assume $K/F$ be a field extension and $A,B \in M_n(F)$ and $P\in GL_n(K)$ such that $PAP^{-1}=B$. Can we find $C\in GL_n(F)$ such that $CAC^{-1}=B$?

Answer 1

Yes, this is true. This is essentially a consequence of the existence and unicity of the Frobenius normal form and from the fact that two matrices over $K$ are conjugated in $M_n(K)$ if and only if they have the same Frobenius normal form.