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Math Help - Matrix Solution

  1. #1
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    Matrix Solution

    I have the matrix H:
    A11=a+bi
    A12=c+di
    A21=-c+di
    A22=a-bi

    I need to show that the equation x^2=-1 has infinitely many solutions in H.
    I ignored a (because the book told me to) and so I have the matrix:
    A11=bi
    A12=c+di
    A21=-c+di
    A22=-bi
    where b^2+c^2+d^2=1.

    I'm just having trouble showing the "infinitely many solutions in H" part.
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  2. #2
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by JoeDardeno23 View Post
    I have the matrix H:
    A11=a+bi
    A12=c+di
    A21=-c+di
    A22=a-bi

    I need to show that the equation x^2=-1 has infinitely many solutions in H.
    I ignored a (because the book told me to) and so I have the matrix:
    A11=bi
    A12=c+di
    A21=-c+di
    A22=-bi
    where b^2+c^2+d^2=1.

    I'm just having trouble showing the "infinitely many solutions in H" part.
    You have one constraint on three variables. So, for example, given a value b = 0, how many solutions can you come up with for c and d? I get an infinite number:

    c = cos \theta and d = sin \theta
    where \theta is a continuous parameter.

    This alone gives you an infinite number of solutions, but you can obviously make a similar kind of argument for any acceptable value of b, that is for |b| \leq 1.

    -Dan
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  3. #3
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    How does this relate back to the matrix though?
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  4. #4
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    Could someone just explain this subtle point?
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  5. #5
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by JoeDardeno23 View Post
    Could someone just explain this subtle point?
    Your matrix is defined by the values b, c, and d by the A11, A12, A21, and A22 relations that you gave in your initial post. The condition that any matrix belonging to the set H be such that x^2 = -1 gives a constraint on the possibilities that b, c, and d can take, but is not so resistrictive as to give a finite set of possibilities.

    -Dan
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