Let consider a real vector space $\mathbb{R}^n$ of dimension $n$, where $\langle \cdot, \cdot\rangle$ and $||\cdot ||$ are the standard inner product and $\ell_2$ indeced norm.

Does there exist a transformation (a non-linear one) $T:\mathbb{R}^n\rightarrow \mathbb{R}^n$ such that it preserves orthogonality **and** amplify/enhance non-orthogonal cosines:

- For any two vectors $x,y\in\mathbb{R}^n$ it holds $\langle x,y\rangle = 0 \iff \langle T(x), T(y)\rangle = 0$, i.e. $T$ just preserves orthogonality
- $\frac{|\langle T(x), T(y)\rangle|}{||T(x)||\cdot||T(y)||} \gg \frac{|\langle x, y\rangle|}{||x||\cdot||y||} $, i.e. we would like to always amplify/enhance cosine between non-orthogonal vectors