Rotation matrix to construct canonical form of a conic
$C:9{x}^{2}+4xy+6{y}^{2}-10=0.$
I've found C is a non-degenerate ellipses (computing the cubic and the quadratic invariant), and then I've studied the characteristic polynomial
$p\left(t\right)=\mathrm{det}\left(\begin{array}{cc}9-t& 2\\ 2& 6-t\\ & \phantom{\rule{0ex}{0ex}}\end{array}\right)$
The eigenvalue are ${t}_{1}=5,{t}_{2}=10$, with associated eigenvectors $(-1,2),(2,1)$. Thus I construct the rotation matrix R by putting in columns the normalized eigenvectors (taking care that $det\left(R\right)=1$):
$R=\frac{1}{\sqrt{5}}\left(\begin{array}{cc}1& 2\\ -2& 1\\ & \phantom{\rule{0ex}{0ex}}\end{array}\right)$
Then $(x,y)}^{t}=R{({x}^{\prime},{y}^{\prime})}^{t$, and after some computations I find the canonical form
$\frac{1}{2}{x}^{\prime 2}+\frac{4}{5}{y}^{\prime 2}=1.$
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