1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: 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


    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper
    Education Advisor

    Can you assume A has an inverse?
  4. Aug 29, 2011 #3
    I figured it out.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook