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Homework Help: Simple Linear Algebra Proof

  1. Feb 25, 2010 #1
    1. The problem statement, all variables and given/known data
    Prove that if A and B are symmetric nxn matrices then AB is symmetric if and only if AB=BA


    2. Relevant equations



    3. The attempt at a solution
    I tried to say AB=(AB)T=BTAT=BA
    But I don't think this is correct.
     
  2. jcsd
  3. Feb 25, 2010 #2

    vela

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    When you have an "if and only if," you need to prove both directions. You have to show:

    1) If AB is symmetric, then AB=BA.
    2) If AB=BA, then AB is symmetric.

    What you have so far is a proof of #1. You assumed AB is symmetric, which means AB=(AB)T, and found AB=BA. So you're half done. You just need to prove #2 now.
     
  4. Feb 25, 2010 #3
    Would the second part be...
    AB=BA=(BA)T=ATBT=AB
     
  5. Feb 25, 2010 #4
    If A=AT and B=BT, and if AB is symmetric,
    then [tex](AB)^T=AB=A^TB^T=(BA)^T[/tex]
    thus for AB to be symmetric, (AB)T=(BA)T therefore AB=BA.
    Almost the same thing you wrote.

    For the second part, if AB=BA, then (AB)T=....
     
    Last edited: Feb 25, 2010
  6. Feb 25, 2010 #5

    vela

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    No, because you don't know that BA=(BA)T.
     
  7. Feb 25, 2010 #6
    Why can AB=(AB)T but we don't know if BA=(BA)T
     
  8. Feb 25, 2010 #7
    Just because [tex] (AB)^T=AB [/tex], doesn't mean [tex] (BA)^T=BA [/tex]
     
  9. Feb 25, 2010 #8

    vela

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    You don't know AB=(AB)T either. That's what you're trying to prove. When you're proving #2, all you know is A and B are symmetric and AB=BA.
     
  10. Feb 25, 2010 #9
    AB=BA=BTAT=(AB)T

    Would this work?
     
  11. Feb 25, 2010 #10

    vela

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    Can you justify each step?
     
  12. Feb 25, 2010 #11
    Well we assume AB=BA. We can say that the next step holds too since we know that A and B are symmetric. The final step is just a property. Is that flawed?
     
  13. Feb 25, 2010 #12

    vela

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    Nope, that's perfect.
     
  14. Feb 25, 2010 #13
    Great thanks again. Whenever you get a chance, I entered another post over on the previous thread. If you can help me verify, it will be great! (even though you have already helped tremendously)
     
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