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Linear algebra proof

  1. Apr 26, 2010 #1
    1. The problem statement, all variables and given/known data

    Let A be a 2x2 symmetric matrix and x be a scalar. Prove that the graph of the quadratic equation (X^T)AX = k is hyperbola if k is non zero and det(A) less than zero

    (T stands for transpose

    2. Relevant equations

    not were to begin
     
  2. jcsd
  3. Apr 27, 2010 #2

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    Well, begin by writing it out!
    Let X= <x, y> and
    [tex]A= \begin{bmatrix} a & b \\ c & d\end{bmatrix}[/tex].

    Then
    [tex]X^TAX= \begin{bmatrix}x & y\end{bmatrix}\begin{bmatrix}a & b \\ c & d\end{bmatrix}\begin{bmatrix} x \\ y\end{bmatrix}=ax^2+ cxy+ bxy+ dy^2= ax^2+ (b+ c)xy+ dy^2[/tex]

    Now, under what conditions on a, b, c, d is [itex]ax^2+ (b+ c)xy+ dy^2[/itex] a hyperbola?
     
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