i am trying to prove that $x^2+xy+2y^2$ is equivalent to $x^2-xy+2y^2$ and that $3x^2+xy+4y^2$ is not equivalent to $3x^2-xy+4y^2$.
As for the first problem, i am trying to find a matrix $M$ with integer coefficients and determinant 1 that takes me from the first quadratic forms to the second one , but i am not able to find it. As for the second problem, i dont know where to start. Can i get some hints please ?
There is a small finite set of matrices that take us from $3 x^2 + xy + 4 y^2$ to $3 x^2 - xy + 4 y^2.$ Call one $$ P = \left( \begin{array}{cc} p & q \\ r & s \end{array} \right) $$ Necessary conditions are that the left column give form value $3,$ and the right column give form value $4.$ That is $$ 3 p^2 + pr + 4 r^2 = 3, $$ $$ 3 q^2 + q s + 4 s^2 = 4. $$ One pair of examples is $$ P = \left( \begin{array}{cc} 1 & 0 \\ 0 & -1 \end{array} \right), $$ $$ P = \left( \begin{array}{cc} -1 & 0 \\ 0 & 1 \end{array} \right). $$ Both of these have determinant $-1,$ the traditional word for such a pair of forms is "opposite." The remaining task for you is to find out any other matrices $P,$ or calculate that there are no others, to conclude that the determinant is always $-1.$ This will not be hard: I think there are very few ways to get $$ 3 p^2 + pr + 4 r^2 = 3, $$ and very few ways to get $$ 3 q^2 + q s + 4 s^2 = 4. $$
Indeed, multiplying by numbers and completing the square two times, we get separate $$ 3 x^2 + x y + 4 y^2 \geq \frac{47}{16} x^2 \approx 2.9375 x^2, $$ $$ 3 x^2 + x y + 4 y^2 \geq \frac{47}{12} y^2 \approx 3.9167 y^2. $$ We can only get $3$ when $y=0.$ We can only get $4$ when $|y| \leq 1$ and $|x| \leq 1,$ very few possibilities to check.