I'm stuck on the following problem, I'd appreciate any sort of hints.
$A,B \in M_n(\mathbb{R})$ and there exists $S \in M_n(\mathbb{C})$ such that $SAS^{-1}=B$. Prove that there exists $T \in M_n(\mathbb{R})$ such that $TAT^{-1}=B$.
What I have tried so far is to assume $S$ to have the form $C+iD$ with $i$ representing the imaginary unit and $S^{-1}$ to have the form $E+iF$ and then what we find is that:
$$CAE-DAF = B$$ $$CAF+DAE = 0$$ $$CE-DF = I_n$$ $$CF+DE = 0$$
such that $C,D,E,F \in M_n(\mathbb{R})$
but I couldn't use these identities to construct an invertible matrix in Real field with required property.
