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  "body": "\n## Summary\n\nPower iteration, or the power method, finds the [[eigenvalue|0BQAA2BK3Q]] of largest absolute value of a matrix and a corresponding [[eigenvector|0N6960W0BV]].\nIt multiplies a starting vector by the matrix again and again and rescales it.\nIt needs only matrix-vector products, so it works for very large sparse matrices, and it is the basis of the PageRank computation.\nIts speed depends on the ratio between the two largest absolute values of the eigenvalues.\n\n## Explanations\n\n### L1 human\n\nApplying a matrix stretches each eigenvector direction by its eigenvalue.\nAfter many applications, the direction with the largest stretch wins over all others.\nFor $A = \\begin{pmatrix} 2 \u0026 1 \\\\ 1 \u0026 2 \\end{pmatrix}$ and $x_0 = (1, 0)$, the vectors $A x_0, A^2 x_0, A^3 x_0$ are $(2, 1)$, $(5, 4)$, $(14, 13)$.\nThey point closer and closer to $(1, 1)$, the eigenvector for the largest eigenvalue $3$.\n",
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