Similitude group for a bilinear form

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.

The symmetry group for a bilinear form is a normal subgroup of the similitude group, and is in fact the kernel of the factor of similitude homomorphism. The image of the homomorphism is termed the factor of similitude group and is a subgroup of the multiplicative group of the field.

There are two special cases:

• When the bilinear form is a symmetric bilinear form, we use the term orthogonal similitude group. The symmetry group in this case is the orthogonal group. When the symmetric bilinear form is the standard dot product, we get the orthogonal similitude group for the standard dot product.
• When the bilinear form is an alternating bilinear form, we use the term symplectic similitude group. The symmetry group in this case is the symplectic group.