# diagonalising matrices over different fields

Say I have two fields $$F$$ and $$F'$$ such that $$F\subseteq F'$$. Say for a matrix $$A \in M_{n}(F)$$ is diagonalisable over $$F$$, then does it automatically mean that $$A$$ is diagonalisable over $$F'$$?

The question I was dealing was a matrix, $$A$$, say, which I found to be diagonalisable over $$\mathbb{R}$$ and so I am thinking then that means $$A$$ is diagonalisable over $$\mathbb{C}$$ right?

This is correct. If you diagonalize over $$F$$, then you will be able to diagonalize over $$F'\supseteq F$$, since you can just use the same operations over $$F'$$ that you have used over $$F$$. The converse is not generally true.
Yes, if $$P^{-1}AP =D$$ where P is a matrix in $$F$$ and $$F\subset F'$$, then $$P$$ is a matrix in $$F'$$ as well.
Thus $$A$$ is diagonalizable in $$F'$$