An extension of the Perron-Frobenius theory to arbitrary matrices and cones

Abstract

We develop the Perron-Frobenius theory using a variational approach and extend it to a set of arbitrary matrices, including those that are neither irreducible nor essentially positive, and do not preserve a cone. We introduce a new concept called a "quasi-eigenvalue of a matrix," which is invariant under orthogonal transformations of variables, and has various useful properties, such as determining the largest value of the real parts of the eigenvalues of a matrix. We extend Weyl's inequality for the eigenvalues to the set of arbitrary matrices and prove the new stability result to the Perron root of irreducible nonnegative matrices under arbitrary perturbations. As well as this, we obtain new types of estimates for the ranges of the sets of eigenvalues and their real parts.