Orthogonal similitude group for a symmetric bilinear form

Definition

Standard definition

Let $k$ be a field, $V$ a (usually finite-dimensional) vector space over $k$, and $b:V\times V\to k$ a bilinear form. The similitude group for $b$ is the group of invertible linear transformations $A:V\to V$ such that there exists a $\lambda$ (dependent on $A$ such that, for all $v,w\in V$, we have:

$\!b(Av,Aw)=\lambda b(v,w)$.

Although the definition does not require $b$ to be nondegenerate, we typically make this assumption.

The value $\lambda$ is termed the factor of similitude or ratio of similitude for $A$. 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.

Alternative definition

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.