Symmetry group for a bilinear form

Definition

Let $k$ be a field, $V$ a (usually finite-dimensional) vector space over $k$, and $b$ a bilinear form on $V$. The symmetry group of $b$ is the group of invertible linear transformations of $V$ satisfying:

$\!b(Av,Aw)=b(v,w)\ \forall \ v,w\in V$.

Typically, we assume the form $b$ to be nondegenerate.

There is special terminology for two special cases:

• The term orthogonal group is used when the bilinear form is symmetric. Two typical examples are the orthogonal group for the standard dot product and split orthogonal group. There may be many other inequivalent symmetric bilinear forms for a given vector space over a field.
• The term symplectic group is used when the bilinear form is alternating. Up to equivalence, there is a unique nondegenerate alternating bilinear form on an even-dimensional vector space over a field (and none for an odd-dimensional vector space).

Relation with other groups

• Similitude group for a bilinear form is the group of all invertible linear transformations that act by a scalar multiple on the bilinear form. The scalar is termed the factor of similitude. There is thus a natural homomorphism from the similitude group to the factor of similitude group and the kernel of that homomorphism is precisely the symmetry group.