$5x^2+8xy+5y^2=\mathbf{x}^TA\mathbf{x}= > (S^T\mathbf{x})^TD(S^T\mathbf{x})=1\left(\frac{x-y}{\sqrt{2}}\right)^2+9\left(\frac{x+y}{\sqrt{2}}\right)^2$ Thus, the equation is that of an ellipse, since it is the sum of two squares. It is tempting to simplify this expression by pulling out factors of 2. However, it is important not to do this. The quantities $c_1=\frac{x-y}{\sqrt{2}},\quad c_2=\frac{x+y}{\sqrt{2}}$ have a geometrical meaning. They determine an orthonormal coordinate system on $\mathbb{R}^2$. In other words, they are obtained from the original coordinates by the application of a rotation (and possibly a reflection). Consequently, one may use the $c_1$ and $c_2$ coordinates to make statements about length and angles (particularly length), which would otherwise be more difficult in a different choice of coordinates (by rescaling them, for instance). For example, the maximum distance from the origin on the ellipse ${c_1}^2 + 9{c_2}^2 = > 1$ occurs when $c_2=0$, so at the points $c1=±1$. Similarly, the minimum distance is where $c_2=±1/3$. (Wikipedia, Principal Axis Theorem)
I am not getting the bold parts. Can anyone explain this? I do understand the parts before the bold ones.
Thanks.
Edit: I do know that this is an ellipse. What I am not sure of is what it means by $c_1$ and $c_2$ being used to determine orthonormal coordinates. (I also do know what orthonormal means, as it appears in elementary linear algebra.)
More specifically, what is "original coordinates", and what rotation + reflection is occuring due to $c_1$ and $c_2$? What would be the center?