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  "body": "\n## Definition\n\nLet $A$ be an $n \\times n$ complex [[matrix|0VD9ZSWWJG]] and let $x_0 \\in \\C^n$ be a nonzero starting vector.\n*Power iteration* is the sequence defined by\n$$x_{k+1} = \\frac{A x_k}{\\norm{A x_k}}, \\qquad k \\ge 0,$$\nas long as $A x_k \\ne 0$, together with the eigenvalue estimates $\\mu_k = x_k^* A x_k / (x_k^* x_k)$.\n\n## Remarks\n\nThe numbers $\\mu_k$ are [[Rayleigh quotients|06WHP8SE98]].\nConvergence: suppose $A$ is diagonalizable with [[eigenvalues|0BQAA2BK3Q]] $\\abs{\\lambda_1} \u003e \\abs{\\lambda_2} \\ge \\cdots \\ge \\abs{\\lambda_n}$, and $x_0$ has a nonzero component along an [[eigenvector|0N6960W0BV]] $v_1$ for $\\lambda_1$ in the eigenbasis.\nThen the direction of $x_k$ tends to that of $v_1$, up to a scalar factor of modulus $1$, with error of order $\\abs{\\lambda_2 / \\lambda_1}^k$, and $\\mu_k \\to \\lambda_1$.\nExample: $A = \\begin{pmatrix} 2 \u0026 1 \\\\ 1 \u0026 2 \\end{pmatrix}$ has eigenvalues $3$ and $1$ with eigenvectors $(1, 1)$ and $(1, -1)$; for $x_0 = (1, 0) = \\tfrac12 (1, 1) + \\tfrac12 (1, -1)$, one gets $A^k x_0 = \\tfrac12 (3^k + 1, 3^k - 1)$, whose direction tends to $(1, 1)$ with error of order $3^{-k}$.\nNon-example of convergence: for $A = \\begin{pmatrix} 1 \u0026 0 \\\\ 0 \u0026 -1 \\end{pmatrix}$, the condition $\\abs{\\lambda_1} \u003e \\abs{\\lambda_2}$ fails, and from $x_0 = (1, 1)$ the iterates alternate between the directions of $(1, -1)$ and $(1, 1)$.\nApplying the method to $(A - \\sigma I)^{-1}$ gives inverse iteration, which finds the eigenvalue closest to a shift $\\sigma$.\n",
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