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Thread: Matrix Conic Equation

  1. #1
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    Matrix Conic Equation

    I need to write this equation here: X1^2+X1X2+X2^2=8

    in the form of X^TAX=8


    Once ive done that i nee to be able to find the matrix Q such that the transformation X=QY diagonalises A and reduces the equation to a standard form in terms of (Y1, Y2)
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  2. #2
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    Quote Originally Posted by sazzy3 View Post
    I need to write this equation here: X1^2+X1X2+X2^2=8

    in the form of X^TAX=8
    You may assume A is symmetric so:

    $\displaystyle \left[\begin{array}{cc}x_1&x_2 \end{array} \right]$ $\displaystyle \left[\begin{array}{cc}a_{1,1}&a_{1,2}\\a_{1,2}&a_{2,2}\ end{array}\right]$ $\displaystyle \left[\begin{array}{c}x_1\\x_2\end{array}\right]=8$

    Now expand the expression on the left and set the coefficient of $\displaystyle x_1^2$ to $\displaystyle 1$, of $\displaystyle x_2^2$ to $\displaystyle 1$ and of $\displaystyle x_1x_2$ also to $\displaystyle 1$ and solve for $\displaystyle a_{1,1}$, $\displaystyle a_{2,2}$ and $\displaystyle a_{2,2}$

    CB
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