Symmetric bilinear form

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.

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