Polar form of a quadratic form

Definition

Let $F$ be a field, let $V$ be a finite-dimensional vector space over $F$, and let $Q$ be a quadratic form on $V$. The polar form of $Q$ is the map $B_Q \colon V \times V \to F$ with $$B_Q(u, v) = Q(u + v) - Q(u) - Q(v).$$

Remarks

$B_Q$ is a symmetric bilinear form, and $B_Q(v, v) = 2 Q(v)$ for all $v$. For $Q(x) = x^T A x$ on $F^n$, $B_Q(x, y) = x^T (A + A^T) y$. Many authors divide by $2$ when the characteristic is not $2$, so that the result $B$ satisfies $B(v, v) = Q(v)$; see the associated form. This construction is the degree $2$ case of the polarisation of homogeneous polynomials.

Explanations

L1 human

The polar form recovers a bilinear form from a quadratic form. For $Q(x) = x^2$ on $\R$, the polar form is $(x + y)^2 - x^2 - y^2 = 2xy$. It is the algebraic form of the polarisation identity for inner products.