Symmetric bilinear form associated with a quadratic form

Definition

Let $F$ be a field of characteristic not $2$, and let $V$ be a finite-dimensional vector space over $F$. Let $Q$ be a quadratic form on $V$. The symmetric bilinear form associated with $Q$ is $$B(u, v) = \tfrac{1}{2} \bigl( Q(u + v) - Q(u) - Q(v) \bigr) \qquad (u, v \in V).$$

Remarks

$B$ is a symmetric bilinear form with $B(v, v) = Q(v)$, and it is the only one. Conversely, each symmetric bilinear form $B$ gives the quadratic form $v \mapsto B(v, v)$; the two constructions are inverse to each other. For $Q(x) = x^T A x$ with $A$ symmetric, $B(x, y) = x^T A y$. It is half of the polar form of $Q$.

Explanations

L1 human

Outside characteristic $2$, quadratic forms and symmetric bilinear forms are the same thing in two forms. From $B$, we get $Q$ by putting the same vector twice. From $Q$, we get $B$ by this formula.