对称双线性形式是在矢量空间上的对称双线性形式。它们在正交极性和二次曲面的研究中非常重要。

设 V 在域 K 上的 n 维矢量空间。映射 是这个空间上的对称双线性形式，如果:

最后两个公理只蕴涵在第一个参数中的线性，但是第一个公理直接蕴涵了在第二个参数中的线性。

A symmetric bilinear form on a vector space is a linear map from two copies of the vector space to the field of scalars such that the order of the two vectors does not affect the value of the map. In other words, it is a bilinear function that maps every pair of elements of the vector space to the underlying field such that for every and in . They are also referred to more briefly as just symmetric forms when "bilinear" is understood.