On polyhedra of Perron-Frobenius eigenvectors

This paper deals with the positive eigenvectors of nonnegative irreducible matrices which are merely characterized by a given upper bound u on their spectral radius and by a given matrix L of lower bounds for their elements. For any such matrix, the normalized positive left [right] eigenvector is shown to belong to the polyhedron the vertices of which are given by the normalized rows [columns] of the matrix $\text{(u}\text{I}\text{?}\text{L}\text{)}{}^{\mathrm{?1}}$. This polyhedron is proven to be also the smallest closed set which is guaranteed to contain the positive left [right] normalized eigenvector; its vertices are therefore the best componentwise bounds one can obtain on the positive eigenvectors of these matrices. A less general result has also been obtained for the symmetrical case, when the matrices are only characterized by a given lower bound l on their spectral radius and by a given matrix U of upper bounds for their elements.