Symmetric bilinear form

In mathematics, a symmetric bilinear form on a vector space is a bilinear 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 ${\displaystyle B}$ that maps every pair ${\displaystyle (u,v)}$ of elements of the vector space ${\displaystyle V}$ to the underlying field such that ${\displaystyle B(u,v)=B(v,u)}$ for every ${\displaystyle u}$ and ${\displaystyle v}$ in ${\displaystyle V}$. They are also referred to more briefly as just symmetric forms when "bilinear" is understood.

Symmetric bilinear forms on finite-dimensional vector spaces precisely correspond to symmetric matrices given a basis for V. Among bilinear forms, the symmetric ones are important because they are the ones for which the vector space admits a particularly simple kind of basis known as an orthogonal basis (at least when the characteristic of the field is not 2).

Given a symmetric bilinear form B, the function q(x) = B(x, x) is the associated quadratic form on the vector space. Moreover, if the characteristic of the field is not 2, B is the unique symmetric bilinear form associated with q.