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I'm trying to show a result from Manfred Stoll's book on Hyperbolic geometry. Consider the function $$M_w(z) = w + \frac{1 - |w|^2}{|w - z|^2}(w - z)$$ where $w,z \in \mathbb{R}^n$ and $|w| < 1$. (I am using different variable names than Stoll). This function generalizes the Möbius transformations on the complex numbers.

Stoll writes that $$|M_w(z)|^2 = \frac{|w - z|^2 + (1 - |w|^2)(1-|z|^2)}{|w - z|^2}$$ How do you show this? I tried computing $\langle M_w(z), M_w(z) \rangle$, but I'm not getting any closer.

This result ultimately shows that $M_w(z)$ maps into the unit ball.

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$$\begin{align} \left|M_w(z)\right|^2 &= |w|^2+\frac{\left(1-|w|^2\right)^2}{|w-z|^4}|w-z|^2+2\frac{1-|w|^2}{|w-z|^2}\;w\cdot(w-z) \\[8pt] |w-z|^2\;\left|M_w(z)\right|^2 &= |w|^2|w-z|^2+\left(1-|w|^2\right)^2+2\left(1-|w|^2\right)\;\left(|w|^2-w\cdot z\right) \\[8pt] &= \phantom{+2}|w|^2\left(|w|^2+|z|^2-2w\cdot z\right)\\ &\phantom{=}+\phantom{2}\left(1-2|w|^2+|w|^4\right)\\ &\phantom{=}+2\left(|w|^2-w\cdot z-|w|^4+|w|^2w\cdot z \right) \\[8pt] &= \left(-2w\cdot z\right)+ \left(1 + |w|^2|z|^2\right)\\[6pt] &= \left(|w|^2+|z|^2-2w\cdot z\right)+ \left(1 -|w|^2-|z|^2+ |w|^2|z|^2\right)\\[6pt] &= |w-z|^2+ \left(1 -|w|^2\right)\left(1-|z|^2\right)\\ \end{align}$$

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