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Prove that A and B are simultaneously diagonalizable.

  1. Aug 28, 2011 #1
    1. The problem statement, all variables and given/known data

    A and B are commuting diagonalizable matrices. Prove that they are simultaneously diagonalizable.

    2. Relevant equations

    AB = BA

    3. The attempt at a solution

    I have what looks like a proof, but I'm not very happy with it. Is there anything wrong here?

    AB = BA
    B = ABA-1


    Every matrix has exactly one jordan form. All diagonal matrices are jordan forms. So there exists a unique marix C such that
    CBC-1 = J(C) where J(C) is C's Jordan form.
    J(C) = CBC-1 = CABA-1C-1 = (CA)B(CA)-1
    As C is unique, C = CA, so A = I
    And CIC-1 = I, which is diagonal.
    So A and B are simultaneously diagonalisable.
     
  2. jcsd
  3. Aug 28, 2011 #2

    vela

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    Can you assume A has an inverse?
     
  4. Aug 29, 2011 #3
    I figured it out.
     
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