Rotate a quadric surface on $\mathbb{R}³$ so the crossed term doesn't appear

Question

Having the general quadric equation on three variables $$Ax²+By²+Cz²+2Dxy+2Eyz+2Fxz+2Gz+2Hy+2Iz+J$$, I need to rotate a quadric surface $$11 x^2 + 9 y^2 + 12 z^2 + (2 \sqrt 3) xy - 72 = 0$$ using the method by eigenvalues, $det\left( \left[ \begin{array}{cc} A&\frac{D}{2}\\ \frac{D}{2}&B \end{array} \right] + \left[ \begin{array}{cc} -\lambda&0\\ 0&-\lambda \end{array} \right]\right) $, but I don't know wich values should I take to fill the associated matrix. I understand by now, if I want to eliminate the $xy$ term it is okay to use this method like it is used on $\mathbb{R}^2$, so If i needed to eliminate an $yz$ term, the values on the associated matrix should be coefficients of $y², z²$ and $yz$?

Answer 1

You're eliminating the $xy$ term, so you should use the coefficients of $x^2$, $xy$, $y^2$.

$$\begin{bmatrix} A & D/2 \\ D/2 & B \end{bmatrix} = \begin{bmatrix} 11 & \sqrt3 \\ \sqrt3 & 9 \end{bmatrix}$$

If you had several mixed terms, like $xy$, $yz$, then you would need to use a $3\times 3$ matrix, and find $3$ eigenvalues.