I have been reading through an article on the method of conjugate gradients (in solving a system $\textbf{A}x=b$). You can find the article here.
In the article, we consider the quadratic form $f(x)= \frac{1}{2}x^\top\textbf{A}x - b^\top x + c$ where $\textbf{A}$ is symmetric, positive-definite. Of course, in the 2D case, the contour lines of $f(x)$ are ellipses. On a graph of these ellipses, it is stated toward the bottom of p. 22 that $\textbf{A}$-orthogonality of two vectors $u$ and $v$ such that $u^\top \textbf{A} v = 0$ implies regular orthogonality ($u^\top v = 0$) when the graph is stretched such that the elliptical contours of $f(x)$ are transformed into circles.
I would love to try and prove this, but I'm not really sure where to start. I feel like a good place to begin would be to analyze the quadratic form to figure out what transformation matrix is necessary to convert its contours into circles.