# Geometric representation of principal axis

I would like to sketch $Q(X)=5x_1^2-4x_1x_2+5x_2^2=48$

So matrix $A=\begin{bmatrix} 5 & -2 \\ -2 & 5 \end{bmatrix}$

The eigenvectors are $\lambda=3, \lambda=7$. So $Q(X)=3y_1^2+7y_1^2=48$, with the orthonormal basis (of the eigenvectors): So matrix $P=\begin{bmatrix} 1/\sqrt{2} & 1/\sqrt{2} \\ 1/\sqrt{2} & -1/\sqrt{2} \end{bmatrix}$ (The eigenvectors are, $\lambda=3: [1, 1]^T, \lambda=7: [1, -1]^T$)

This is where I am stuck, how can I find the correct new axis of this representation?

you have the eigenvectors which give the principal axes of the ellipse, i.e. the lines of symmetry are $y=x$ and $y=-x$ ![1]: http://i.stack.imgur.com/hDDbR.png][1]
• Thank you for your answer! This makes it clearer. I however don't see why this elipse 'rotates' at $[3, 3]$ and $[-3, -3]$. Could you explain me how that works? – Holograph Jun 23 '15 at 12:00