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  "body": "\n## Definition\n\nLet $U \\subseteq \\mathbb{R}^n$ be open, let $f \\colon U \\to \\mathbb{R}$, and let $a \\in U$ be a point at which all second [[partial derivatives|02RSXVM1ZN]] of $f$ exist.\nThe Hessian matrix of $f$ at $a$ is the $n \\times n$ matrix\n$$H_f(a) = \\left( \\frac{\\partial^2 f}{\\partial x_i \\, \\partial x_j}(a) \\right)_{1 \\le i, j \\le n},$$\nwhose entry in row $i$ and column $j$ is $\\partial^2 f / \\partial x_i \\, \\partial x_j$ at $a$.\nIt is also written $\\nabla^2 f(a)$ or $D^2 f(a)$.\n\n## Remarks\n\nIf the second partial derivatives of $f$ are continuous near $a$, then $H_f(a)$ is a symmetric matrix, by [[Schwarz's theorem|0NNQY8WCW6]].\nFor $f(x, y) = x^2 + 3 x y - y^2$, the Hessian matrix at each point is $\\begin{pmatrix} 2 \u0026 3 \\\\ 3 \u0026 -2 \\end{pmatrix}$.\nThe Hessian matrix is the Jacobian matrix of the gradient map $x \\mapsto \\nabla f(x)$, when this map is differentiable.\nAt a critical point, the signs of the eigenvalues of $H_f(a)$ decide the [[second derivative test|0GYDV9FE98]].\n\n## Explanations\n\n### L1 human\n\nThe gradient tells how a function tilts; the Hessian tells how it bends.\nFor a bowl like $x^2 + y^2$ the Hessian is $2 I$, and both directions bend upward. For a saddle like $x^2 - y^2$ one direction bends up and the other bends down.\n",
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