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Homework Help: Similar matricies

  1. Jun 17, 2010 #1
    1. The problem statement, all variables and given/known data

    Let A and B be 2x2 real matricies, and suppose there exists an invertible complex 2x2 matrix P such that B = [P^(-1)]AP.

    Show that there exists a real invertible 2x2 matrix Q such that B = [Q^(-1)]AQ.

    2. Relevant equations
    A and B are similar when thought of as complex matricies, so they represent the same linear transformation on C2 for appropriately chosen bases, and share many other properties:

    same trace, same determinant, same characteristic equation , same eigenvalues.




    3. The attempt at a solution
    If I take Q = (1/2)(P + P bar) (the "real part" of P),
    then I can show QB = AQ, and so B = [Q^(-1)]AQ if Q is invertible, but this Q may not be invertible.

    I also noticed that each of A and B may be triangularized (since each of A and B has an eigenvalue), but I don't know where to go from there...
     
  2. jcsd
  3. Jun 18, 2010 #2

    lanedance

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    Homework Helper

    does it help to know as A & B are real, then:
    [tex] B* = B [/tex]
    [tex] A* = A[/tex]

    which gives
    [tex]B = (P^{-1})^*AP^* = P^{-1}AP [/tex]
     
  4. Jun 26, 2010 #3
    I don't know if this helps, since P* may not be real.
     
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