Symmetric Bilinear Form

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

(1)

which satisfies for all (Conrad n.d., p. 1). For the matrix descriptions and classifications below, assume that has finite dimension .

For example, on , if is an symmetric matrix, then

(2)

is a symmetric bilinear form. Consider

(3)

then

(4)

When has field characteristic different from 2, 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 symmetric bilinear form is called nondegenerate if its kernel is zero. That is, if for every nonzero , 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 an invertible matrix such that

(7)

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

Over , a nondegenerate symmetric bilinear form can be diagonalized by choosing an orthogonal basis and rescaling its vectors, 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.

Over , 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.

Over , 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. Over other fields, the classification depends on the field. For instance, if the field has field characteristic 2, then it is not possible to divide by 2 since . The polarization correspondence above does not apply in field 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.

For a nondegenerate form, a change of basis multiplies by . The coset of in is a well-defined invariant of , called the discriminant. For real forms, it is represented by 1 or . For , the discriminant has a nonzero rational number representative where and are squarefree. On a vector space of fixed dimension over a finite field of odd field characteristic, a symmetric bilinear form is determined by its rank and the discriminant of its nondegenerate part. This classification does not apply in field characteristic 2.

A nondegenerate 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.