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  "body": "\n## Definition\n\nLet $V$ be a real vector space of finite dimension $n$, and let $Q$ be a [[quadratic form|0RBQS1JJCR]] on $V$.\nChoose a basis in which $Q(x_1 e_1 + \\cdots + x_n e_n) = a_1 x_1^2 + \\cdots + a_n x_n^2$.\nLet $p$, $q$ and $r$ be the numbers of indices $i$ with $a_i \u003e 0$, $a_i \u003c 0$ and $a_i = 0$.\nThe *signature* of $Q$ is the triple $(p, q, r)$.\n\n## Remarks\n\nBy [[Sylvester's law of inertia|0WFED3K3DQ]], the triple does not depend on the choice of the basis.\nFor a symmetric matrix $A$ of $Q$, the numbers $p$, $q$ and $r$ count the positive, negative and zero eigenvalues of $A$, with multiplicity.\nConventions vary: some authors write only $(p, q)$, or the number $p - q$.\nThe Minkowski form $x_1^2 - x_2^2 - x_3^2 - x_4^2$ on $\\R^4$ has $p = 1$, $q = 3$ and $r = 0$.\n\n## Explanations\n\n### L1 human\n\nThe signature counts the plus signs, the minus signs and the zeros in a real quadratic form, after diagonalisation.\nIt is the complete invariant of a real quadratic form.\nTwo real quadratic forms are equivalent exactly when they have the same signature.\n",
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