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Prove: (AB)*=A*B*

  1. Sep 20, 2015 #1

    RJLiberator

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    1. The problem statement, all variables and given/known data
    Show that (AB)*=A*B*


    2. Relevant equations

    * = complex conjugate.
    A,B = Matrices, A is an nxm matrix and B is a mxl matrix.

    3. The attempt at a solution

    Okay, last problem on this large, lovely homework assignment.

    I feel like theres two general ways this homework has gone.

    Either 1) Use summation notation to look at elements of the matrices and find that the components on each side are equal.

    OR
    2) Use complex properties to prove the statements.

    I want to use route 2) as it is easier and more beautiful.

    But I've been staring at this, seemingly simple, statement for a while now and can't check my first move.
    It seems like an obvious statement.

    If I let C = AB, and say c is within the complex numbers.
    Then C* = the conjugate of C.

    But this isn't what I want to prove, methinks.
     
  2. jcsd
  3. Sep 20, 2015 #2

    fzero

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    Perhaps it would help to show first that you can write a complex matrix as a linear combination of real matrices, in analogy with what we can do with numbers.
     
  4. Sep 21, 2015 #3

    Dick

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    Now you should write the matrix product as a summation. And then use that ##(ab)^*=a^*b^*## for complex numbers.
     
  5. Sep 21, 2015 #4

    RJLiberator

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    Oh man, that's an easy one!

    You write the summation out, and use the fact that you are now dealing with real numbers and can use that property and boom!

    How do you write summation notation in this scenario?

    Here's what I did:

    1. [itex]\left( \sum_{k=0}^m(a_{ij}b_{ij})^*\right)[/itex]
    2. [itex]\left( \sum_{k=0}^ma_{ij}^*b_{ij}^*\right)[/itex]
    3. =(a*b*)_ij
    4. = A*B*

    Step 1 is component notation
    Step 2 is property of complex numbers
    Step 3+4 bring it home.
     
  6. Sep 21, 2015 #5

    Dick

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    You are summing on an index, k, that isn't even in the expression you are summing. Look up the right way to express matrix multiplication in index form, ok?
     
  7. Sep 21, 2015 #6

    RJLiberator

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    Er, for some reason I did it right in my homework, but wrong on here. Probably too much focus on the latex.

    K=1 to m.
    a_ik
    b_kj
     
  8. Sep 21, 2015 #7

    Dick

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    That's better.
     
  9. Sep 21, 2015 #8

    Fredrik

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    Right, matrix multiplication is defined by ##(AB)_{ij}=\sum_k A_{ik}B_{kj}## and the adjoint is defined by ##(C^*)_{ij}=(C_{ji})^*##. (The asterisk on the right denotes complex conjugation. You may prefer the notation ##\overline C_{ji}##).

    I'm not sure what most linear algebra books call the matrix that I called the adjoint, but I hope they don't call it the "complex conjugate", because that would be very misleading. C* denotes the transpose of the matrix that you get when you take the complex conjugate of each element of a matrix C.
     
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