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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?

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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.

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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 $F'$

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