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Math Help - Matrices and Complex Numbers

  1. #1
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    Matrices and Complex Numbers

    Hey,
    So I'm studying the links between complex numbers and matrices at the moment, and have been using the matrix
    0 1
    -1 0
    to express the complex number i, (which represents the square root of -1).
    I was wondering if there are any other ways to express i as a matrix??
    Thankyou!! :)
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  2. #2
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    Re: Matrices and Complex Numbers

    No, that is not the only way to represent "i". Another obvious choice would be \begin{pmatrix}0 & -1 \\ 1 & 0\end{pmatrix} but there are many others.

    I presume that you are representing the real number, a, by the matrix \begin{pmatrix}a & 0 \\  0 & a\end{pmatrix} so that -1 is represented as \begin{pmatrix}-1 & 0 \\ 0 & -1\end{pmatrix}. So if [tex]\begin{pmatrix} a & b \\ c & d \end{pmatrix}[tex], where a, b, c, and d are real numbers, represents the "square root of negative one", we must have
    \begin{pmatrix}a & b \\ c & d \end{pmatrix}^2= \begin{pmatrix}a^2+ bc & ab+ bd \\ ac+ cd & bc+ d^2\end{pmatrix}= \begin{pmatrix}-1 & 0 \\ 0 & -1\end{pmatrix}

    and so must have a^2+ bc= -1, ab+ bd= 0, ac+ cd= 0, and  bc+ d^2= -1.

    From ab+ bd= b(a+ d)= 0, we must have either b= 0 or a+ d= 0. If b= 0 then a^2+ bc= -1 becomes a^2= -1 which is impossible so d= -a.
    From ac+ cd= c(a+ d)= 0 we get the same thing. The other two equations then become a^2+ bc= -1 so that bc= -a^2- 1.

    The simplest thing to do is to take a= d= 0 and either b= 1, c= -1 or b=-1, c= 1 but there many other choices.
    Thanks from NJoyce
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  3. #3
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    Re: Matrices and Complex Numbers

    Thankyou for your help!
    I'm a bit confused about how there can be any other "i" values apart from those two though, as if you put whole numbers in the leading diagonal then it would take the place of the real values of a complex number. And if you use numbers higher than 1 it no longer results in a -1 scalar matrix. So what other choices could there be?
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  4. #4
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    Re: Matrices and Complex Numbers

    Thankyou for your help, what you said makes sense, but I've tried and tried and I can't find any other values for i apart from those two, not without including the letter i in one of the elements. You said there are lots of ways, can you tell me of any other ways to express it??
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  5. #5
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    Re: Matrices and Complex Numbers

    Nevermind I've got it after all
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  6. #6
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    Re: Matrices and Complex Numbers

    Suppose we did this:

    $a+bi \mapsto \begin{bmatrix}a+b&2b\\-b&a-b \end{bmatrix}$

    It is evident that this map preserves complex addition: what about multiplication?

    Well:

    $\begin{bmatrix}a+b&2b\\-b&a-b \end{bmatrix}\begin{bmatrix}c+d&2d\\-d&c-d \end{bmatrix} = \begin{bmatrix}(ac-bd)+(ad+bc)&2(ad+bc)\\-(ad+bc)&(ac-bd)-(ad+bc) \end{bmatrix}$

    which is the image of the complex number:

    $(ac-bd) + (ad+bc)i = (a+bi)(c+di)$

    So here we have the matrix:

    $\begin{bmatrix}1&2\\-1&-1 \end{bmatrix}$

    playing the role of "$i$".

    The moral of the story is: we have SEVERAL "copies" of the complex numbers in the ring of 2x2 real matrices, and there is no way to say which copy is the "actual" one.
    Thanks from topsquark
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