Subsections

• Defining a Quadratic Form
• Properties of the Form
• Structure of Inner Product Spaces

A bilinear form may be defined on a vector space when it is constructed using the VectorSpace-constructor.

[Future release] AssignForm(V, F) : ModTupFld, AlgMatElt ->

Suppose that the vector space V is a subspace of K^((n)). Given a matrix F belonging to Mat_K(n) that defines a bilinear form on V with respect to the current basis for V, assign F as the bilinear form associated with V.

The discriminant of the quadratic form defined by the symmetric matrix Q.

Return the bilinear form on V as a matrix.

Convert the quadratic form f into a symmetric matrix.

Convert the symmetric matrix a belonging to Mat_K(n) into a polynomial, where a is interpreted as the matrix of a quadratic form.

The rank of the bilinear form defined by the form matrix F.

The rank of the quadratic form defined by the symmetric matrix F.

The Witt index of the quadratic form defined by the symmetric matrix Q.

An isotropic vector in V relative to the bilinear form currently defined on V.

Construct a maximal isotropic subspace in V relative to the bilinear form currently defined on V.

Construct an orthogonal basis for the vector space V with respect to the bilinear form currently defined on V.

The orthogonal complement of the subspace U in the vector space V with respect to the bilinear form currently defined on V.