$M_n(F)$ is set of all $n\times n$ matrices over field $F$. Suppose we have $A,B\in M_n(\mathbb{R})$ and invertible matrix $P\in M_n(\mathbb{C})$ such that $P^{-1}AP=B$. Show that an invertible matrix $Q\in M_n(\mathbb{R})$ exists such $Q^{-1}AQ=B$.
I searched for the answer and found some answers to it: 1, 2, 3, 4, etc. Some other answers suggested to use Jordan or rational canonical forms that I don't want to use (I'm not allowed). Answer 3 is comprehensible but is using determinant that I'm also not allowed to show a matrix is invertible using determinants (but I can use other theorems).
Answers in 2 and 4, both claim that we can write entries of $P$ using a basis of the vector space over $\mathbb{R}$. I can't understand this part. Entries of $P$ are in $\mathbb{C}$, so how can we claim that we can write them using elements in $\mathbb{R}$? Maybe there's something I can't understand.
Anyways, any new methods to proove the statement are appreciated. Also any explanations about those answers I linked.
