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  "body": "\n## Summary\n\nAn eigenvalue algorithm is a numerical procedure that computes [[eigenvalues|0BQAA2BK3Q]], and often [[eigenvectors|0N6960W0BV]], of a square matrix.\nBy the [[Abel–Ruffini theorem|163A2WQQJX]], the eigenvalues of a general matrix of size at least $5$ cannot be written with radicals.\nSo such algorithms are iterative and produce approximations.\nThey are central in numerical linear algebra, with uses from vibration analysis to search engines and quantum chemistry.\n\n## Explanations\n\n### L1 human\n\nAn eigenvector of a matrix is a direction that the matrix only stretches, and the eigenvalue is the stretch factor.\nFor a $2 \\times 2$ matrix one can find the eigenvalues by solving a quadratic equation, but for large matrices no formula exists.\nEigenvalue algorithms instead improve a guess step by step until it is accurate enough.\nThe simplest one, [[power iteration|H4MQD841BR]], multiplies a vector by the matrix again and again; the vector turns toward the direction of the largest stretch.\nFor example, repeatedly applying $\\begin{pmatrix} 2 \u0026 0 \\\\ 0 \u0026 1 \\end{pmatrix}$ to $(1, 1)$ gives $(2^k, 1)$, which points more and more along $(1, 0)$, the eigenvector for eigenvalue $2$.\n",
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