Linear algebra proof

  • Thread starter blackblanx
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  • #1
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Homework Statement



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

Homework Equations



not were to begin
 

Answers and Replies

  • #2
HallsofIvy
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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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