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Finding crossing regions between two quadractics

  1. Oct 16, 2015 #1
    Given two quadratics of the form $f(x) = x'Q_1x$ and $g(x) = x'Q_2x$ and assuming $Q_1$ and $Q_2$ are negative definite matrices, how can I find the lines that are formed by their crossing? here I'm assuming $x \in \mathbb{R}^2$.

    4EIjohE.png
    up5lbdC.png

    I'm interested in finding vectors $w_1$ and $w_2$ that are parallel to the blue dashed line. I also outlined the eigenvectors of $Q_1-Q-2$, they seem to point exact in the middle of each region.
     
  2. jcsd
  3. Oct 16, 2015 #2

    andrewkirk

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    Write x as (x y)' and label the eight (known) entries of the two matrices, then write out ## x'(Q_1-Q_2)x=0## as an expansion that will be of the form ##\alpha x^2+\beta xy+\gamma y^2##. Then solve for ##\frac{x}{y}##.
     
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