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  "body": "\n## Summary\n\nA condition number measures how much the output of a problem can change, relative to the size of a small relative change in its input.\nFor an [[invertible matrix|0EB9N0S33C]] $A$ and a [[matrix norm|54T6Y6DSY4]], it is $\\kappa(A) = \\norm{A} \\norm{A^{-1}}$.\nIt bounds the relative error in the solution of $Ax = b$ caused by a relative error in $b$.\nA problem with a large condition number is ill-conditioned: rounding errors in the data can be strongly amplified, whatever algorithm is used.\n\n## Explanations\n\n### L1 human\n\nSome problems are sensitive: a tiny change in the data gives a big change in the answer.\nThe condition number puts a number on this sensitivity.\nFor a linear system $Ax = b$, a condition number of $1000$ means that a $0.1\\%$ error in $b$ can cause up to a $100\\%$ error in $x$.\nFor example, the diagonal matrix with entries $1$ and $0.001$ has condition number $1000$ in the Euclidean norm.\n",
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