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  "body": "\n## Definition\n\nLet $A$ be an [[invertible|0EB9N0S33C]] real $n \\times n$ matrix, and consider the [[linear system|0GT4WWGWH2]] $A x = b$.\nA *preconditioner* for $A$ is an invertible $n \\times n$ matrix $P$, used through the map $r \\mapsto P^{-1} r$.\nThe *left preconditioned* system is $P^{-1} A x = P^{-1} b$.\nThe *right preconditioned* system is $A P^{-1} y = b$ with $x = P^{-1} y$.\nIf $P = P_1 P_2$, the *split preconditioned* system is $P_1^{-1} A P_2^{-1} y = P_1^{-1} b$ with $x = P_2^{-1} y$.\n\n## Remarks\n\nAll three systems have the same solution $x$ as $A x = b$.\nThe definition is formal; a preconditioner is useful when two informal requirements hold.\nFirst, $P^{-1} r$ is cheap to compute for any $r$.\nSecond, $P^{-1} A$ is close to the identity, for example it has a small [[condition number|AB3S6MHJ8X]] or few clusters of [[eigenvalues|0BQAA2BK3Q]].\nThe two extremes are $P = I$, which is cheap but changes nothing, and $P = A$, which is ideal but as costly as the original problem.\nExample: the Jacobi preconditioner $P = \\operatorname{diag}(a_{11}, \\dots, a_{nn})$ is invertible when all diagonal entries are nonzero.\nFor $A$ with rows $(100, 1)$ and $(1, 1)$, the ratio of the extreme eigenvalues of $A$ is about $101$.\nThe matrix $P^{-1} A$ has rows $(1, 0.01)$ and $(1, 1)$ and eigenvalues $0.9$ and $1.1$, with ratio about $1.2$.\nNon-example: the matrix $P$ with rows $(1, 1)$ and $(1, 1)$ is singular, so it is not a preconditioner.\nFor a symmetric [[positive definite|0KX66CZMBB]] $A$ one takes $P$ symmetric positive definite, so that methods for symmetric systems still apply.\n",
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