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  "body": "\n## Definition\n\nLet $n \\ge 1$ and let $\\mathbb{K}$ be $\\R$ or $\\C$.\nAn *eigenvalue algorithm* is an algorithm that takes as input a [[matrix|0VD9ZSWWJG]] $A \\in \\mathbb{K}^{n \\times n}$ (and, for iterative algorithms, a tolerance $\\varepsilon \u003e 0$) and returns:\n\n1. numbers $\\tilde\\lambda_1, \\dots, \\tilde\\lambda_k \\in \\C$, with $1 \\le k \\le n$, that approximate [[eigenvalues|0BQAA2BK3Q]] $\\lambda_1, \\dots, \\lambda_k$ of $A$, counted with algebraic multiplicity, to within the tolerance;\n2. optionally, vectors $\\tilde v_1, \\dots, \\tilde v_k$ that approximate corresponding [[eigenvectors|0N6960W0BV]] or bases of invariant subspaces.\n\nIt computes *all* eigenvalues when $k = n$, and *some* eigenvalues otherwise, for example those of largest modulus or those closest to a given shift.\n\n## Remarks\n\nExample: the [[QR algorithm|B6F8YWHYZ5]] reduces $A$ to Hessenberg form and applies shifted QR steps until the matrix is nearly upper triangular.\nIts diagonal then approximates all $n$ eigenvalues, so it is an eigenvalue algorithm with $k = n$.\nNon-example: an algorithm that only computes the coefficients of the [[characteristic polynomial|0VSSPAFM2C]] does not return eigenvalues, so clause 1 fails.\nFinding the roots is a further step, and it is badly conditioned in general.\nThe eigenvalues of a general matrix with $n \\ge 5$ are not expressible by radicals in its entries.\nSo every eigenvalue algorithm that works for all such matrices is iterative and stops at a tolerance.\nAccuracy is usually measured by backward error: the output should be the exact eigenvalues of a matrix $A + E$ with $\\norm{E}$ small compared with $\\norm{A}$.\n\n## Explanations\n\n### L1 human\n\nAn eigenvalue algorithm is any recipe that, given a square matrix, outputs its eigenvalues (and maybe eigenvectors) up to a chosen accuracy.\nSome recipes find all of them, others find only a few that one asks for.\n",
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