• A
  • A
  • A
  • ABC
  • ABC
  • ABC
  • А
  • А
  • А
  • А
  • А
Regular version of the site

Article

How to make the Perron eigenvector simple

Calcolo. 2019. Vol. 56. No. 2. P. 1-11.

Multiple Perron eigenvectors of non-negative matrices occur in applications, where

they often become a source of trouble. A usual way to avoid it and to make the

Perron eigenvector simple is a regularization of matrix: an initial non-negative matrix

A is replaced by A + "M, where M is a strictly positive matrix and " > 0 is small.

However, this operation is numerically unstable and may lead to a signicant increase

of the Perron eigenvalue, especially in high dimensions. We dene a selected Perron

eigenvector of A as the limit of normalized Perron eigenvectors of the regularizations

A + "M as " ! 0. It is shown that if the matrix M is rank-one, then the limit

eigenvector can be found by an explicit formula and, moreover, is eciently computed

by the power method. The role of the rank-one condition is explained.