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  "body": "\n## Summary\n\nThe Schur complement of an invertible block $A$ in a [[block matrix|0ASDFEQS6P]] $\\begin{pmatrix} A \u0026 B \\\\ C \u0026 D \\end{pmatrix}$ is the matrix $D - C A^{-1} B$.\nIt is what remains of the block $D$ after block Gaussian elimination removes the block $C$.\nIt gives the [[determinant via the Schur complement|0KA7A4ENVJ]], formulas for the inverse of a block matrix, and tests for positive definiteness.\n\n## Explanations\n\n### L1 human\n\nThink of a system of two vector equations in two unknown vectors $x$ and $y$.\nIf you solve the first equation for $x$ and put the result into the second, the matrix in front of $y$ is the Schur complement.\nFor numbers, take $\\begin{pmatrix} a \u0026 b \\\\ c \u0026 d \\end{pmatrix}$ with $a \\ne 0$.\nThe Schur complement of $a$ is the number $d - cb/a$.\nThen the determinant $ad - bc$ equals $a \\cdot (d - cb/a)$.\n",
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