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Given a field $F$ and a subfield $K$ of $F$. Let $A$, $B$ be $n\times n$ matrices such that all the entries of $A$ and $B$ are in $K$. Is it true that if $A$ is similar to $B$ in $F^{n\times n}$ then they are similar in $K^{n\times n}$?

Any help ... thanks!

  • 9
    The answer is yes. This follows from the classification of finitely generated modules over a principal ideal domain. That's classical. Briefly, $A$ and $B$ are similar iff, for all $k \le n$, the gcd of the order $k$ minors of $A-X$ and $B-X$ coincide, and this doesn't depend on the field. (Here $X$ is an indeterminate.) I can look for references, if you want.2011-08-13
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    I should have said that the gcd's are computed in $K[X]$ or in $F[X]$, and the point is you obviously get the same gcd's. Also, $A-X$ is the matrix whose determinant is the characteristic polynomial (in case you're used to other notation). - That's in all the standard algebra books. I can look for online references.2011-08-13
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    Sorry, but I didn't get the idea! Could you please be more clear, or is there any other method?2011-08-13
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    It’s a little hard to explain all this stuff in a comment. Again, I can give you references to books; I can look for online references; or you can wait for an answer. I’m pretty sure people will be happy to answer. If you have a precise question on what I said, I can try to answer it. I tried to summarize in a few lines what takes several pages in books.2011-08-13
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    Here is a first [reference](http://eom.springer.de/e/e035300.htm).2011-08-13
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    You can use the rational canonical form, but one has to be a bit careful (because the uniqueness of the rational canonical form depends on the monic irreducible factors one chooses).2011-08-13
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    @ArturoMagidin: How so "the uniqueness of the rational canonical form depends on the monic irreducible factors one chooses"? Uniqueness is uniqueness; one can show explicitly that distinct rational canonical forms are not similar. Also the rational canonical form does not use or depend on any irreducible factors; this is why it is _rational_ (independent of field extensions).2013-12-13

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