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  "body": "\n## Definition\n\nLet $A \\in \\C^{n \\times n}$, let $\\Lambda(A)$ be its set of [[eigenvalues|0BQAA2BK3Q]], and let $\\varepsilon \u003e 0$.\nLet $\\norm{\\cdot}$ be the Euclidean norm on $\\C^n$ with its [[operator norm|06EKW1AKXF]] on matrices.\nThe *$\\varepsilon$-pseudospectrum* of $A$ is\n$$\\Lambda_\\varepsilon(A) = \\Lambda(A) \\cup \\set{z \\in \\C \\setminus \\Lambda(A) : \\norm{(zI - A)^{-1}} \u003e 1/\\varepsilon}.$$\n\n## Remarks\n\nThe following sets are equal to $\\Lambda_\\varepsilon(A)$:\n\n1. $\\set{z \\in \\C : z \\in \\Lambda(A + E) \\text{ for some } E \\in \\C^{n \\times n} \\text{ with } \\norm{E} \u003c \\varepsilon}$;\n2. $\\set{z \\in \\C : \\norm{(zI - A) v} \u003c \\varepsilon \\text{ for some unit vector } v}$;\n3. $\\set{z \\in \\C : \\sigma_{\\min}(zI - A) \u003c \\varepsilon}$, where $\\sigma_{\\min}$ is the smallest [[singular value|09VMDSRXWJ]].\n\nThe convention $\\norm{(zI - A)^{-1}} = \\infty$ for $z \\in \\Lambda(A)$ makes the definition read $\\norm{(zI - A)^{-1}} \u003e 1/\\varepsilon$ for all $z$.\n\nExample: if $A$ is normal, then $\\norm{(zI - A)^{-1}} = 1/\\mathrm{dist}(z, \\Lambda(A))$, so $\\Lambda_\\varepsilon(A)$ is the union of the open discs of radius $\\varepsilon$ about the eigenvalues.\nExample: for $J = \\begin{pmatrix} 0 \u0026 1 \\\\ 0 \u0026 0 \\end{pmatrix}$ and $z \\ne 0$, $(zI - J)^{-1} = \\begin{pmatrix} 1/z \u0026 1/z^2 \\\\ 0 \u0026 1/z \\end{pmatrix}$, whose norm is at least $1/\\abs{z}^2$; so every $z$ with $0 \u003c \\abs{z} \u003c \\sqrt{\\varepsilon}$ lies in $\\Lambda_\\varepsilon(J)$.\nNon-example: for $A = 0 \\in \\C^{1 \\times 1}$ and $\\varepsilon = 0.1$, the point $z = 0.2$ is not in $\\Lambda_\\varepsilon(A)$, since $\\norm{(zI - A)^{-1}} = 5 \\le 10$.\nThe sets are open, they increase with $\\varepsilon$, and their intersection over all $\\varepsilon \u003e 0$ is $\\Lambda(A)$.\nThe same definition makes sense for a bounded operator on a Banach space, with its [[spectrum|BMKATJ802A]] in place of $\\Lambda(A)$.\n\n## Explanations\n\n### L1 human\n\nA point is in the $\\varepsilon$-pseudospectrum when it is \"almost an eigenvalue\": some vector is mapped by $A - z$ to a vector shorter than $\\varepsilon$.\nThese sets are drawn as contour plots of the resolvent norm in the complex plane.\n",
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