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  "body": "\n## Definition\n\nLet $\\mathbb{K}$ be $\\R$ or $\\C$, let $n \\ge 1$, and let $\\norm{\\cdot}$ be a submultiplicative [[matrix norm|54T6Y6DSY4]] on $\\mathbb{K}^{n \\times n}$.\nLet $A$ be an [[invertible|0EB9N0S33C]] $n \\times n$ matrix over $\\mathbb{K}$.\nThe *condition number* of $A$ with respect to $\\norm{\\cdot}$ is\n$$\\kappa(A) = \\norm{A} \\, \\norm{A^{-1}}.$$\n\n## Remarks\n\nBy convention, $\\kappa(A) = \\infty$ when $A$ is not invertible.\nIf $\\norm{\\cdot}$ is an [[operator norm|06EKW1AKXF]], then $\\kappa(A) \\ge \\norm{A A^{-1}} = \\norm{I} = 1$.\nFor the operator norm of the Euclidean norm, $\\kappa_2(A) = \\sigma_{\\max} / \\sigma_{\\min}$, the ratio of the largest and smallest [[singular values|09VMDSRXWJ]].\nIf $Ax = b$ with $b \\ne 0$ and $A(x + \\delta x) = b + \\delta b$, then $\\norm{\\delta x} / \\norm{x} \\le \\kappa(A) \\, \\norm{\\delta b} / \\norm{b}$ for the operator norm of a vector norm.\nExample: $A = \\begin{pmatrix} 1 \u0026 0 \\\\ 0 \u0026 10^{-3} \\end{pmatrix}$ has $\\norm{A}_2 = 1$ and $\\norm{A^{-1}}_2 = 10^{3}$, so $\\kappa_2(A) = 10^3$.\nNon-example: $\\begin{pmatrix} 1 \u0026 1 \\\\ 1 \u0026 1 \\end{pmatrix}$ is not invertible, so the formula does not apply.\n",
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