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I ran into an interesting result which showed that if $A$ is an orthogonal matrix then $$\left\vert\left\vert Ax \right\vert\right\vert = \left\vert\left\vert x \right\vert\right\vert$$ where $x \in \mathbb{R}^n$.

So now I am wondering if all orthogonal mappings are also conformal mappings.

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okay nevermind,

I just saw this paper

EDIT

Okay,

Now that I have more time, I will explain:

A conformal matrix $M = R \cdot O$ where $R$ is positive definite and $O$ is orthogonal.

The paper continues on:

If $M$ is an invertible matrix then the image by $M$ of the unit sphere is an ellipse whose principal axes have length $2\cdot \lambda_j$ where the $\lambda_j$ are the eigenvalues of $R$ in the polar decomposition $M = R\cdot O$. The direction of the axes are the corresponding eigendirections.

The image is a sphere if an only if $R = cI$ with $c > 0$ if and only if $M^t \cdot M = c^2\cdot I$.

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