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Math Help - quadratic equations in matrix form.

  1. #1
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    quadratic equations in matrix form.

    Hi

    I'm sure this question will seem seem very trivial, but I was wondering if you could help me out.

    Although I can see that quadratic equations such as

    5x^2 -6xy+5y^2=8 can be shown as the matrix (xy)(5 -3/-3 5)=8 (Using the divide symbol to indicate row down.)

    Similarly

    7x^2-12xy-2y^2=10 would be (x y)(7 -6 /-6 -2)

    but I was stuck when trying to convert the quadratic equation x^2 + x-8 +5xy -6y +2y^2=0 into matrix form.

    If someone could shed some knowledge on the subject I would be deeply appreciative.

    Thanks

    Matt
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  2. #2
    Moo
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    Hello,

    That's something about quadratic forms. If I'm not mistaking, the third one isn't one, so you can't find the matrix form of this equation...

    Also, don't forget \begin{pmatrix} x\\ y\end{pmatrix} on the right side of the 2x2 matrix...
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  3. #3
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    Hi there, thanks for the response.

    Then reason I asked was because I am trying to transform the last equation so that it can be expressed in the form
    AX^2 + BY^2=1 and was trying to do this using eigen analysis, and I'm not sure if this is possible without matrices?
    Do you know of any other ways I could set about doing this?

    Thanks
    Matt
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  4. #4
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    Actually, you can do it, but it's a bit tricky, and it has to be with shifted variables. You'd like to write your equation this way:

    (x+a)^{2}+5(x+a)(y+b)+2(y+b)^{2}=C.

    Multiplying this out and comparing coefficients with your original equation lead to the following three equations:

    2a+5b=1

    5a+4b=-6

    C=8+a^{2}+5ab+2b^{2}.

    You can solve this system rather straight-forwardly. Then you can write your quadratic form as the following:

    \begin{bmatrix}x+a&y+b\end{bmatrix}\begin{bmatrix}  1 &5/2\\<br />
5/2 &\sqrt{2}\end{bmatrix}\begin{bmatrix}x+a\\ y+b\end{bmatrix}=C.

    This may or may not be allowed, depending on your definitions, as Moo pointed out. But this computation can still be done.
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