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Basic Linear Algebra Proof

  1. Jul 18, 2009 #1
    1. The problem statement, all variables and given/known data

    Prove the following:

    For every n × n complex matrices A and B, [tex](\alpha AB)^*=\bar{\alpha }B^*A^*[/tex].

    2. Relevant equations

    None

    3. The attempt at a solution

    Okay, I'm just getting started on this problem. All the ideas I have come up with so far involve using two "test" matrices. The problem with this is that it doesn't prove it for any n × n matrix. Does this matter?
     
  2. jcsd
  3. Jul 18, 2009 #2

    Office_Shredder

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    Ok, start small. Can you prove


    [tex]
    (\alpha A)^*=\bar{\alpha }A^*
    [/tex]

    If you can, then you can just focus on proving that A and B swap like that
     
  4. Jul 18, 2009 #3
    Hmm...

    This feels like trying to prove 1+1=2. It just is! I'm still working on it...
     
  5. Jul 18, 2009 #4
    Do you think this is a sufficient proof?

    [tex]
    (\alpha AB)^*=\bar{\alpha }\overline{AB}=\bar{\alpha }\left(\bar{A}\right)\left(\bar{B}\right)=\bar{\alpha }\left(\left(\bar{A}\right)^T\right)^T\left(\left(\bar{B}\right)^T\right)^T=\bar{\alpha }\left(A^*\right)^T\left(B^*\right)^T=\bar{\alpha }\left(B^*A^*\right)^T
    [/tex]
     
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