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  "body": "\n## Summary\n\nEigenvalue perturbation studies how the [[eigenvalues|0BQAA2BK3Q]] and [[eigenvectors|0N6960W0BV]] of a matrix change when the matrix is changed by a small amount.\nFor a simple eigenvalue, the change is smooth, and its first-order term is given by left and right eigenvectors.\nBounds such as [[Weyl's inequality|RACSNQXDQV]] and the [[Bauer–Fike theorem|AK9J4RNZ51]] control the change without expansions.\nThe subject matters for numerical computation, where matrices are known only up to rounding errors.\nIt also matters in physics, where it is the base of perturbation theory.\n\n## Explanations\n\n### L1 human\n\nChange the entries of a matrix a little and ask how its eigenvalues move.\nFor a symmetric matrix they move by at most the size of the change.\nFor example, the eigenvalues of $\\begin{pmatrix} 1 \u0026 \\varepsilon \\\\ \\varepsilon \u0026 2 \\end{pmatrix}$ are $1 - \\varepsilon^2 + \\dots$ and $2 + \\varepsilon^2 + \\dots$, so they barely move.\nFor some non-symmetric matrices a tiny change can move the eigenvalues a lot.\n",
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