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Unitary matrix problem

  1. Mar 10, 2007 #1
    1. The problem statement, all variables and given/known data

    I am to prove that adjoint of(AB)= adjoint of B times adjoint of A

    2. Relevant equations



    3. The attempt at a solution

    I satrted this as U=adjoint of AB
    u_ik=sum(j)[(a_ij)*(b_jk)]
    I know then,I may take tarnspose of both sides so that we have:[(u_ki)~]=sum(j){[(b_kj)~][(a_ji)~]}
    then [U~]=[B~][A~]
    Then,we are done.But,this proves for real elements.I am not sure that this proves also for imaginary elements...
    Please help.
     
  2. jcsd
  3. Mar 10, 2007 #2

    nrqed

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    The proof works exactlythe same way, you just have to include a complex conjugate of the elements when you take the adjoint. Taking the complex conjugate does not change anything to the indices, so the proof still works...you just have complex conjugates everywhere.
     
  4. Mar 10, 2007 #3
    OK,I thought of this possibility as I am not using any extra property of complex matrices.
    Do I need to write (a_ij)* in those cases?
     
  5. Mar 10, 2007 #4

    Dick

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    I think you guys are making this too complicated. By definition:

    <x,Ay>=<adjoint(A)x,y> for all x,y.

    So:

    <x,ABy>=<x,A(By)>=<adjoint(A)x,By>=<adjoint(B)*adjoint(A)x,y> for all x,y.
     
    Last edited: Mar 10, 2007
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