Must $S$ be a unitary matrix?

Question

Let $S\in\mathcal{M}_{n\times n}$ invertible matrix, and let $x,y\in\mathbb{R}^n$ prove or give a counterexample that :

$$S \textrm { is unitary }\iff\frac{\|Sx-Sy\|}{\|Sx\|}=\frac{\|x-y\|}{\|x\|} $$

Answer 1

Of course the direction $\Longrightarrow$ is trivially true.

In the converse direction, consider matrices of the form $\lambda\cdot \text{Id}$, for some constant $|\lambda| \neq 1$.

Answer 2

The answer is no for this you can take $S_\alpha=\alpha T$ for all $\alpha\in\mathbb{C}$ an $T$ is any unitary matrix then $S_\alpha$ is unitary if and only if $|\alpha|=1$ but : $$ \frac{\|Sx-Sy\|}{\|Sx\|}=\frac{|\alpha|\|T(x-y)\|}{|\alpha|\|Tx\|}=\frac{\|x-y\|}{\|x\|} $$