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Math Help - What comes first.. multiplication or transpose?

  1. #1
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    What comes first.. multiplication or transpose?

    What would I do first in a problem like this:

    -2(AB)^T

    Obviously first I work work AB. But then would I do the tranpose before multiple -2 in, or multiply -2 in before doing the transpose? I know they end up in the same answer but what way is correct? Which first?
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  2. #2
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    Re: What comes first.. multiplication or transpose?

    Quote Originally Posted by kmjt View Post
    What would I do first in a problem like this:

    -2(AB)^T

    Obviously first I work work AB. But then would I do the tranpose before multiple -2 in, or multiply -2 in before doing the transpose? I know they end up in the same answer but what way is correct? Which first?
    Transpose, what you have means -2 times the transpose of AB.

    CB
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  3. #3
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    Re: What comes first.. multiplication or transpose?

    if (-2)[(AB)^T] = (-2AB)^T, then why are you worrying about this? the map A-->A^T is linear.
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  4. #4
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    Re: What comes first.. multiplication or transpose?

    I may be asking a bit confusing. I am wondering would I multiply the -2 into the outcome of AB before determing the transpose, or other way around?
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  5. #5
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    Re: What comes first.. multiplication or transpose?

    if you get the same answer either way, then what is your concern?

    suppose you multiply AB. now you have a matrix, let's call it M. if we multiply by -2 first, we have -2M. what is the ij-th entry of M? let's call it (gee, i am SO original) m_ij.

    so when we multiply by -2, we now have -2(m_ij). now we take the transpose, and we have -2(m_ji) as our ij-th entry (it got multiplied by -2 as well, everything did).

    ok, now let's do it "the other way". we have m_ij, and after we take the transpose, we have the ij-th entry is: m_ji. now we multiply by -2, and we get: -2(m_ji).

    it looks to me like we get the same matrix either way.

    i'll say it again, the transpose map is LINEAR. that means, in particular, that (cA)^T = c(A^T).
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