Power method

The power method is an iterative approximative method for calculating the eigenvectors of a matrix.

The method is mostly used for calculating the eigenvector of the largest eigenvalue for large matrices. (Google uses it to calculate the page rank of documents in their search engine[1].)

The idea begins by taking any vector $x_{0}$. At the kth stage, compute $x_{k}=Ax_{k-1}$. Then $x_{k}$ converges usually to the eigenvector of the largest eigenvalue.

The method can also be used to calculate the spectral radius of a matrix by computing the Rayleigh quotient

${\frac {x_{k}^{\top }Ax_{k}}{x_{k}^{\top }x_{k}}}={\frac {x_{k+1}^{\top }x_{k}}{x_{k}^{\top }x_{k}}}.$

See also

• Eigenvalue algorithm which describes the algorithm in more detail

External links

• Power method on www.math.buffalo.edu.
• Power method on www.math.gatech.edu