Rotate the axes of to get rid of the cross product term of 9x^2+y^2+6xy=4? I believe I have to find the symmetric coefficient matrix and diagonalize it, but I'm not sure how to do that.

Question

Rotate the axes of to get rid of the cross product term of 9x^2+y^2+6xy=4?  I believe I have to find the symmetric coefficient matrix and diagonalize it, but I'm not sure how to do that.

anonymous · Answer

Well:
$$9x^2+y^2+6xy=4 \rightarrow \vec{r}^T M \vec{r}=4 \implies (x,y)\left[\begin{matrix}9 & 3 \ 3 & 1\end{matrix}\right] \left(\begin{matrix}x \ y\end{matrix}\right)=4$$

anonymous · Answer

So you want to diagnolize M I take it.

anonymous · Answer

Yes!  Is that the correct way to solve the problem?

anonymous · Answer

Yes, you will need to diagnolize it. This is a better explanation: http://ltcconline.net/greenl/courses/203/MatrixOnVectors/conics.htm

anonymous · Answer

Thank you!