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: 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


    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


    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    Ok, start small. Can you prove

    (\alpha A)^*=\bar{\alpha }A^*

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

    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?

    (\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
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook