If I have a conic equation $$ 5x^2 - 4xy + 8y^2 = 36 $$ and $ \left[\begin{array}{cc} 5 & -2\\ -2 & 8 \end{array}\right] $ in matrix form, whose eigenvalues are 4 and 9, how would I rotate the coordinate system such that there is no cross term. Also, how do I identify the type of conic section by doing this?
Thanks.
Every rotation in 2d is determined by a fixed point and a rotational angle. Given the eigenvalues and the center (note that no term of first order exists and hence the origin), the conic equation in new coordinate system $(x',y')$ shall be $4x'^2+9y'^2=C$. The equation obviously describes a ellipse (since $4,9$ are different and positive) or two lines according to $C$. We now know the fixed point is the origin, then it's routine to determine the rotational angle $a$ by identifying the original equation with the new one plugged into $x'=x\cos a-y\sin a$, $y'=x\sin a+y\cos a$.