Orthogonal similitude group for a symmetric bilinear form
Let be a field, a (usually finite-dimensional) vector space over , and a bilinear form. The similitude group for is the group of invertible linear transformations such that there exists a (dependent on such that, for all , we have:
Although the definition does not require to be nondegenerate, we typically make this assumption.
The value is termed the factor of similitude or ratio of similitude for . This gives a homomorphism from the similitude group to the multiplicative group of the field.
This is a special case of the similitude group for a bilinear form.
In the case where the symmetric bilinear form is nondegenerate, the orthogonal similitude group can also be characterized as the normalizer in the general linear group of the orthogonal group for the same bilinear form.
A special case of this is the orthogonal similitude group for the standard dot product.