A symmetric bilinear form is a bilinear form on a vector space that is symmetric. More simply, it is a scheme (equivalently, a function) which maps a pair of elements from the some vector space to its underlying field, it is called symmetric because the order of the elements into the function does not change the element of the field to which it maps. Symmetric bilinear forms are of great importance in the study of orthogonal polarity and quadrics.
They are also more briefly referred to as just symmetric forms when "bilinear" is understood. They are closely related to quadratic forms; for the details of the distinction between the two, see ε-quadratic forms.
Read more about Symmetric Bilinear Form: Definition, Matrix Representation, Orthogonality and Singularity, Orthogonal Basis, Orthogonal Polarities
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