What do you call a scaled orthogonal map?

What 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" aka. "angle-preserving" feels rather close, but I believe these terms are 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" are what I am looking for, but these only capture 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.

• Please let me know if this should be CommunityWiki or is even too opinion based for that! May 23 at 17:39
• Similarity transformation comes close. May 23 at 17:54
• I think "conformal" is the most commonly used term, and it's made clear that only linear transformations are being considered. May 23 at 18:05
• linear conformal ............... May 23 at 19:37
• In Italian I've heard the word "roto-omotetia" to denote the composition of a rotation and a homotethy (mostly in Euclidean affine geometry though). "Roto-homothety" returns a few results in English, too (and apparently not from Italian authors). May 23 at 22:32

Wikipedia suggests "conformal orthogonal group" for the group of all such maps; see the articles

The same term is used in Magma handbook:

http://magma.maths.usyd.edu.au/magma/handbook/text/317

and in quite a few other reputable places, e.g.

https://people.maths.bris.ac.uk/~matyd/GroupNames/linear.html

so it appear that "conformal orthogonal transformation" is, even if slightly tautological when taken literally, the way to go.

"Orthogonal similitude" would be consistent with a very-common use of "symplectic similitude" (in automorphic forms and repn theory) for $$g\in GL_n$$ such that $$g^\top J g=\nu(g)\cdot J$$ for skew-symmetric matrix $$J$$.

And $$GSp(J)$$ is the "symplectic similitude group", and $$GO(S)$$ (with $$S$$ the non-degenerate quadratic form) is the orthogonal similitude group...