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  "body": "\n## Summary\n\nThe QR algorithm computes all the [[eigenvalues|0BQAA2BK3Q]] of a square matrix.\nIt repeatedly factors the matrix as $QR$ by the [[QR decomposition|0ERWG4NXQ9]] and multiplies the factors in the reverse order, $RQ$.\nEach step is a [[similarity|0E89HQATM8]] by an [[orthogonal matrix|0W02QESPJQ]], so the eigenvalues do not change.\nThe matrices tend to an [[upper triangular|0N4XY7Z3NN]] form with the eigenvalues on the diagonal.\nWith shifts and a preliminary reduction to [[Hessenberg form|YCEEWJTX56]], it is the standard method for dense eigenvalue problems.\n\n## Explanations\n\n### L1 human\n\nThe algorithm takes a matrix and splits it into an orthogonal part $Q$, which preserves lengths and angles, and a triangular part $R$.\nThen it puts them back together the other way round.\nThe new matrix has the same eigenvalues as the old one.\nRepeating this pushes the entries below the diagonal towards zero, so the eigenvalues appear on the diagonal.\nFor $A = \\begin{pmatrix} 2 \u0026 1 \\\\ 1 \u0026 2 \\end{pmatrix}$, one step gives $\\begin{pmatrix} 2.8 \u0026 0.6 \\\\ 0.6 \u0026 1.2 \\end{pmatrix}$, already closer to the eigenvalues $3$ and $1$ on the diagonal.\n",
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