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This is a exercise question from Denis Serre, Matrices: Theory and Applications.

Let $M$ and $N$ be two similar matrices in the field $K$. Let $k$ be the subfield spanned by the entries of $M$ and $N$. We have to show that $M$ is similar to $N$ in the subfield $k$ also.

Any hint or complete solution will be really helpful.

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See a similar question spanned from the past over the same field of research. – user1551 Dec 13 '12 at 11:57
up vote 3 down vote accepted

$M $ is similar to its rational canonical form $R_M$ over the sub field $F$

$N$ is similar to its rational canonical form $R_N$ over the sub field $F$

now the rational canonical form is unique $\implies$$R_M$ is the rational canonical form of $M$ and $R_N$ for $N$ over the field $K$

now as $M \sim N$ over $K$ $\implies R_M $ =$R_N$ hence $M$ and $N$ have the same rational form over the subfield $F$ $\implies M \sim N$

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This certainly is in the right direction, but "the rational canonical form is unique" really should be "the rational canonical form does not change when extending the field", and this is not entirely obvious from its definition (also depends on what is the definition), so this point might be elucidated a bit. Also the subfield is calld $k$ in the question, and the last sentence seems to be missing a final "over $k$. – Marc van Leeuwen Dec 13 '12 at 12:39

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