{
  "aliases": [
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  "body": "\n## Summary\n\nLet $A$ be a symmetric $n \\times n$ matrix, and let $Q(x) = x^{\\mathsf{T}} A x$ be its real quadratic form on $\\mathbb{R}^n$. The form $Q$ is positive definite if $Q(x) \u003e 0$ for all $x \\ne 0$. It is negative definite if $Q(x) \u003c 0$ for all $x \\ne 0$.\nIt is positive or negative semidefinite if $Q(x) \\ge 0$ or $Q(x) \\le 0$ for all $x$, and indefinite if it takes both positive and negative values.\n\n## Remarks\n\nThe form $Q$ is positive definite if and only if all eigenvalues of $A$ are positive. It is indefinite if and only if $A$ has a positive and a negative eigenvalue.\nOn $\\mathbb{R}^2$, $x_1^2 + x_2^2$ is positive definite, $x_1^2$ is positive semidefinite but not definite, and $x_1^2 - x_2^2$ is indefinite.\nIn the [[second derivative test|0GYDV9FE98]], the matrix $A$ is the Hessian matrix of a function at a critical point.\n\n## Explanations\n\n### L1 human\n\nA positive definite form is a bowl: it is zero at the origin and positive everywhere else.\nAn indefinite form is a saddle: it goes up in some directions and down in others.\n",
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