I am reading this monograph by Jonathan Richard Shewchuk on conjugate gradient descent method. Call two vector $a$ and $b$ A-orthonomal if $a^TAb=0$, where $A$ is a symmetric positive definite matrix. In page 28-29 a geometric intuition of this is provided without explanation. It says
Figure 22(a) shows what A-orthogonal vectors look like. Imagine if this article were printed on bubble gum, and you grabbed Figure 22(a) by the ends and stretched it until the ellipses appeared circular. The vectors would then appear orthogonal, as in Figure 22(b).
All this images are in that monograph. I fail to understand this. Can anyone please provide some rigorous explanation for this.
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