How to make the Perron eigenvector simple
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.