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I would love your help with this:

Let $F$ be a field and $F\subset K$ a field extension.

Let $A,B\in M_{5}(F)$.

How does one prove that if $A$ and $B$ are similar in $F$, then they are similar in $K$?

Thank you.

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Write out what it means for A,B to be similar as matrices over F. –  hardmath Apr 29 '11 at 22:46
The converse is not quite as trivial. For that you can use the fact that if a system of linear equations over $F$ has a solution over $K$, then it has a solution over $F$. –  Robert Israel Apr 29 '11 at 22:53

1 Answer 1

up vote 3 down vote accepted

Let $L$ be a field. We say that two matrices $X,Y\in M_n(L)$ are similar when there is an invertible $P\in M_n(L)$ such that $$Y=P^{-1}XP.$$ Because $F\subset K$, we can choose to view a matrix with entries in $F$ as also being a matrix with entries in $K$.

Thus, if $A$ and $B$ are similar as matrices in $F$, that is, if $$B=P^{-1}AP$$ where $P\in M_5(F)$ is an invertible matrix, then we also have that $A$ and $B$ are similar as matrices in $K$, that is, $$B=Q^{-1}AQ$$ where $Q\in M_5(K)$ is an invertible matrix - namely, we just take $Q=P$, viewed as a matrix with entries in $K$.

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