Connection Between Bilinear Form and Symmetrie Axes of an Ellipse Given the ellipse $(x,y)\in \mathbb{R}: x^2-2xy+2y^2=1$, we can define it by the symmetric bilinear form $$\sigma(v,w)=v^tAw,\quad A:= \begin{bmatrix}1 & -1\\-1 & 2 \end{bmatrix}$$
as the set of $x \in \mathbb{R^2}:\sigma(x,x)=1$
By experimentation, I have found that the axes of the ellipse are the eigenspaces of the matrix A 
$$v_1:=\begin{bmatrix}\frac{1-\sqrt 5}{2}\\1\end{bmatrix}$$
$$v_2:=\begin{bmatrix}\frac{1+\sqrt 5}{2}\\1\end{bmatrix}$$
Why is this true and how can I formalize this result?

That statement is known as the principal axis theorem and you will find a formal statement here.
