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  "body": "\n## Summary\n\nThe $\\varepsilon$-pseudospectrum of a square matrix is a set of complex numbers.\nIt consists of the numbers that become [[eigenvalues|0BQAA2BK3Q]] after some perturbation of the matrix of norm less than $\\varepsilon$.\nEquivalently, it is the region where the norm of the [[resolvent|J5YNNKSB9P]] $(zI - A)^{-1}$ exceeds $1/\\varepsilon$.\nFor non-normal matrices it can be much larger than the set of eigenvalues.\nIt then explains transient growth, slow convergence of iterative methods and sensitivity of computed eigenvalues, which the eigenvalues alone do not show.\n\n## Explanations\n\n### L1 human\n\nEigenvalues of some matrices are very fragile: a tiny change of the entries moves them far away.\nThe pseudospectrum draws, for a given tolerance $\\varepsilon$, all the places the eigenvalues could move to under a change of size $\\varepsilon$.\nFor a symmetric matrix this is just a disc of radius $\\varepsilon$ around each eigenvalue.\nFor the matrix $\\begin{pmatrix} 0 \u0026 1 \\\\ 0 \u0026 0 \\end{pmatrix}$, whose only eigenvalue is $0$, it contains a disc of radius $\\sqrt{\\varepsilon}$, which is far larger than $\\varepsilon$ when $\\varepsilon$ is small.\n",
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