# Similar matrices over R, Q

1. Dec 9, 2008

### PingPong

1. The problem statement, all variables and given/known data
Show that two matrices that have rational entries and are similar over the reals are also similar over the rationals. (Hint: Consider the polynomials from the rational canonical form over Q. What happens when we consider A as a real matrix?

2. Relevant equations
Rational canonical form (we've mostly been dealing with Q) is unique

3. The attempt at a solution

I've not gotten much. I've noticed that if f is the minimal polynomial of A over Q, g is the minimal polynomial of B over Q, and h is the minimal polynomial of A and B over R (it's the same since they're similar) then h divides both f and g over R. I wanted to somehow show that f divides g and reverse the argument to get g dividing f, but I haven't been able to do that.

In terms of the hint, if we look at the polynomials from the RCF of A over Q, they just get split when we consider consider A over R. A friend of mine suggested that this with the fact that the RCF is unique might help, though I'm not seeing how.

Thanks for any help that you can give.

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