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  "body": "\n## Definition\n\nLet $V$ be a finite-dimensional real vector space, and let $Q$ be a [[quadratic form|0RBQS1JJCR]] on $V$.\nThe form $Q$ is *positive definite* when\n$$Q(v) \u003e 0 \\qquad \\text{for all } v \\in V \\text{ with } v \\ne 0.$$\n\n## Remarks\n\n$Q(x) = x^T A x$ with $A$ symmetric is positive definite if and only if $A$ is a [[positive definite matrix|0KX66CZMBB]].\nEquivalently, the signature of $Q$ is $(n, 0, 0)$, with $n = \\dim V$.\nThe form $x^2 + y^2$ is positive definite on $\\R^2$; the forms $x^2 - y^2$ and $x^2$ are not.\n$Q$ is positive definite if and only if its associated symmetric bilinear form is an inner product.\n\n## Explanations\n\n### L1 human\n\nA positive definite quadratic form is positive everywhere except at the origin.\nIts graph is a bowl with the lowest point at the origin.\nSuch forms describe squared lengths for different inner products.\n",
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