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:

  1. 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
  2. $\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

This isn't possible, even if you let $T$ be non-linear. You can see this in $\mathbb{R}^2$ as follows, but the intuition generalizes straightforwardly for larger $n$:

Let $\{e_1,e_2\}$ be the standard basis of $\mathbb{R}^2$ and let $v := (1,1)/\sqrt{2}$. Then $\langle T(e_1), T(e_2) \rangle = 0$. You want $T(v)$ to have an angle of strictly less than $\pi/4$ with each of $T(e_1)$ and $T(e_2)$, which is impossible (angles satisfy the triangle inequality in $\mathbb{R}^2$, so (angle between $T(e_1)$ and $T(v)$) $+$ (angle between $T(v)$ and $T(e_2)$) $\geq$ (angle between $T(e_1)$ and $T(e_2)$) $= \pi/2$.

  • $\begingroup$ Hi Nathaniel, good point, thank you for answer. $\endgroup$ – Xorwell Dec 23 '13 at 11:58

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