Symmetric Bilinear Form

A symmetric bilinear form on a vector space is a bilinear function

(1)

which satisfies .

For example, if is a symmetric matrix, then

(2)

is a symmetric bilinear form. Consider

(3)

then

(4)

A quadratic form may also be labeled , because quadratic forms are in a one-to-one correspondence with symmetric bilinear forms. Note that is a quadratic form. If is a quadratic form then it defines a symmetric bilinear form by

(5)

The kernel, or radical, of a symmetric bilinear form is the set of vectors

(6)

A quadratic form is called nondegenerate if its kernel is zero. That is, if for all , there is a with . The rank of is the rank of the matrix .

The form is diagonalized if there is a basis , called an orthogonal basis, such that is a diagonal matrix. Alternatively, there is a matrix such that

(7)

is a diagonal quadratic form. The th column of the matrix is the vector .

A nondegenerate symmetric bilinear form can be diagonalized, using Gram-Schmidt orthonormalization to find the , so that the diagonal matrix has entries either 1 or . If there are 1s and s, then is said to have matrix signature . Real nondegenerate symmetric bilinear forms are classified by their signature, in the sense that given two vector spaces with forms of signature , there is an isomorphism of the vector spaces which takes one form to the other.

A symmetric bilinear form with , for all nonzero , is called positive definite. For example, the usual inner product is positive definite. A positive definite form has signature . A negative definite form is the negative of a positive form and has signature . If the form is neither positive definite nor negative definite, then there must exist vectors such that , called isotropic vectors.

A general symmetric bilinear form can be diagonalized with diagonal entries 1, , or 0, because the form is always nondegenerate on the quotient vector space . If is a complex vector space, then a symmetric bilinear form can be diagonalized to have entries 1 or 0. For other fields, there are more symmetric bilinear forms than in the real or complex case. For instance, if the field has field characteristic 2, then it is not possible to divide by 2 since . Hence there is no correspondence between quadratic forms and symmetric bilinear forms in characteristic 2.

The symmetric bilinear forms on a vector space, whose field is not real, have been classified for some fields. There are also theorems about symmetric bilinear forms on free Abelian groups, for example .

A symmetric bilinear form corresponds to a matrix by giving a basis and setting . Two symmetric bilinear forms are considered equivalent if a change of basis takes one to the other. Hence, , where is any invertible matrix. Therefore, the rank of the symmetric bilinear form is an invariant.

Also, can change by . The coset of in is a well-defined invariant of , called the discriminant. For real forms, it is either 1 or -1. For , the discriminant can be any rational number where and are squarefree. A symmetric bilinear form on a finite field is determined by its rank and its discriminant.

A symmetric bilinear form on the p-adic numbers is characterized by its rank, discriminant, and another invariant . Given a basis , orthogonal for , define , then

(8)

where is the Hilbert symbol.

Two symmetric bilinear forms are equivalent on the rationals iff they are equivalent in every as well as the reals (also called .) The data in can be thought of as "local" information, which can be patched together to yield "global" information in . So rational forms have a countable number of distinct invariants, three for every prime number, and two for the reals.