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Math Help - unitary matrix problem

  1. #1
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    unitary matrix problem

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

    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.
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by kolahalb View Post
    I am to prove that adjoint of(AB)= adjoint of B times adjoint of A

    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.
    Let U=AB, then:

    U(i,j) = sum(k) A(i,k)B(k,j)

    adj(U)(i,j) = conj(U(j,i))=conj(sum(k) A(j,k)B(k,i)

    ..............= sum(k) conj(A(j,k)) conj(B(k,i)

    ..............= sum(k) conj(B'(i,k)) conj(A'(k,j) = (adj(B) adj(A))(i,j)

    hence adj(U) = adj(B) adj(A)

    RonL
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