Power method

The power method is a 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 does for instance use it to calculate the page rank of documents in their search engine[1].

The idea is to take any vector $x_{0}$. At the k-th 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