How do you call a linear map of the form $\alpha X$, where $\alpha\in\Bbb R$ and $X\in\mathrm O(V)$ is an orthogonal map ($V$ being some linear space with inner product)? Are there established names, historical names, some naming attempts that haven't caught on?

- "[Conformal](https://en.wikipedia.org/wiki/Conformal_map)" aka. "angle-preserving" feels rather close, but I believe this terms is more commonly used in the sense of "locally angle-preserving" (i.e. it is not implicitly understood to be linear). Also, $\alpha=0$ is explicitly allowed in my context, which is not quite angle-preserving.

- I first thought "[homotheties](https://en.wikipedia.org/wiki/Homothety)" are what I am looking for, but these only captures the scaling part, not the rotation part.

- Roto-scaling or scale-rotation is apparently also already taken and is more general than what I need (see the comment by Carlo).

At the risk of letting this become too "opinion-based", let me also say that I am open for suggestions.