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  "body": "\n## Statement\n\nLet $F$ be a field of characteristic not $2$, and let $V$ be a vector space over $F$ of finite dimension $n$.\nLet $Q$ be a [[quadratic form|0RBQS1JJCR]] on $V$.\nThen $V$ has a basis $(e_1, \\dots, e_n)$ and there are $a_1, \\dots, a_n \\in F$ such that, for all $x_1, \\dots, x_n \\in F$,\n$$Q(x_1 e_1 + \\cdots + x_n e_n) = a_1 x_1^2 + a_2 x_2^2 + \\cdots + a_n x_n^2.$$\n\n## Remarks\n\nThe basis is [[orthogonal|01TXN7ETH2]] for the associated symmetric bilinear form, and $a_i = Q(e_i)$.\nIn matrix terms, each symmetric matrix $A$ over $F$ has an invertible $P$ such that $P^T A P$ is diagonal.\nThe numbers $a_i$ are not unique; for example, each $a_i$ can be multiplied by a nonzero square.\nOver $\\R$, the orthogonal diagonalisation of the [[principal axis theorem|0PR5P9Z4KM]] is a special case.\n\n## Explanations\n\n### L1 human\n\nEvery quadratic form becomes a sum of squares with coefficients after a good change of variables.\nFor example, $x^2 + 2xy = (x + y)^2 - y^2$.\nWith $u = x + y$ and $v = y$, the form becomes $u^2 - v^2$.\n",
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