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  "body": "\n## Definition\n\nLet $F$ be a field, let $V$ be a finite-dimensional vector space over $F$, and let $Q$ be a [[quadratic form|0RBQS1JJCR]] on $V$.\nThe *polar form* of $Q$ is the map $B_Q \\colon V \\times V \\to F$ with\n$$B_Q(u, v) = Q(u + v) - Q(u) - Q(v).$$\n\n## Remarks\n\n$B_Q$ is a [[symmetric bilinear form|0AXZNE5XNH]], and $B_Q(v, v) = 2 Q(v)$ for all $v$.\nFor $Q(x) = x^T A x$ on $F^n$, $B_Q(x, y) = x^T (A + A^T) y$.\nMany 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|0KRVRV5V80]].\nThis construction is the degree $2$ case of the polarisation of homogeneous polynomials.\n\n## Explanations\n\n### L1 human\n\nThe polar form recovers a bilinear form from a quadratic form.\nFor $Q(x) = x^2$ on $\\R$, the polar form is $(x + y)^2 - x^2 - y^2 = 2xy$.\nIt is the algebraic form of the [[polarisation identity|05FKHRT0B4]] for inner products.\n",
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