let $X,Y$ be two vector spaces furnished with inner products. Now consider a linear map $L:X\rightarrow Y$ with the following additional property: there exists a constant $C>0$ with
$$\langle Lv,Lw \rangle=C\cdot \langle v,w \rangle, \forall v,w\in X.$$
I am wondering if linear maps with this property do have a special name? This property does look so natural, but I could not find anything in the internet and in books about linear algebra.
Surely, one can call it a "conformal" map, since it is a conformal map in the sense of Riemannian geometry, when we think about the spaces $X,Y$ as Riemannian manifolds. But I would like to know if there is a name for it in the realm of linear algebra.
Thanks in advance