If I have a quadratic in the form $q(x y)=x^2+4xy+y^2$
and I want to express it in the form of normal coordinates.
This is what I've tried so far
$A$ s.t $(x y)A(x y)^T$ which is
A=$\begin{pmatrix} 1 &2 \\2 &1 \end{pmatrix}$
I find eigenvalues $\lambda =-1 ,\lambda =3 $ and eigenvectors
$v1=\begin{pmatrix}-1\\ 1\end{pmatrix}$ and $v2=\begin{pmatrix}1\\ 1\end{pmatrix}$
if I normalise my eigenvectors I get $v1=\frac{1}{\sqrt{2}}\begin{pmatrix}-1\\ 1\end{pmatrix}$ and $v2=\frac{1}{\sqrt{2}}\begin{pmatrix}1\\ 1\end{pmatrix}$
my question is how would I express my original $q(xy)$ in terms of its normal coordinates.