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I need to show the binary quadratic forms $5x^2+xy+y^2$ and $x^2-xy+5y^2$ are equivalent. We've only touched on quadric forms, and the only definition I have for "equivalence" is that one can be transformed into the other via a substitution:

$x = px' +qy', \hspace{15mm}y=rx'+sy'$

with $ps-qr=1$. How can I find such a substitution? Or is there a way to do this without actually having to find the substitution itself?

Thank you guys for any insight into this. This is getting beyond what I can keep in my head

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    Great link, thank you!2012-11-14

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Sure. Take the "Hessian" matrix of the first form as $ H = \left( \begin{array}{cc} 10 & 1 \\ 1 & 2 \end{array} \right) $ Now take the matrix $P \in SL_2 \mathbb Z$ given by $ P = \left( \begin{array}{cc} 0 & -1 \\ 1 & 0 \end{array} \right) $ and calculate $ P^T H P $

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    Ohhh..That makes so much more sense lol. Much appreciated!2012-11-13