I just saw this paper
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
$M^t \cdot M = c^2\cdot I$.