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Similar matrices over R, Q

  1. Dec 9, 2008 #1
    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.
  2. jcsd
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