How to make the Perron eigenvector simple

Abstract

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 $$A + \varepsilon M$$ , where M is a strictly positive matrix and $$\varepsilon > 0$$ is small. However, this operation is numerically unstable and may lead to a significant increase of the Perron eigenvalue, especially in high dimensions. We define a selected Perron eigenvector of A as the limit of normalized Perron eigenvectors of the regularizations $$A + \varepsilon M$$ as $$\varepsilon \rightarrow 0$$ . 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 efficiently computed by the power method. The role of the rank-one condition is explained.