A matrix is said to have the Perron-Frobenius property if it has a positive dominant eigenvalue that corresponds to a nonnegative eigenvector. Matrices having this and similar properties are studied in this paper. Characterizations of collections of such matrices are given in terms of the spectral projector. Some combinatorial, spectral, and topological properties of such matrices are presented...

The sign-real and the sign-complex spectral radius, also called the generalized spectral radius, proved to be an interesting generalization of the classical Perron-Frobenius theory (for nonnegative matrices) to general real and to general complex matrices, respectively. Especially the generalization of the well-known Collatz-Wielandt max-min characterization shows one of the many one-to-one cor...

Fluctuations of the mean field of a globally coupled dynamical systems are discussed. The origin of hidden coherence is related with the instability of the fixed point solution of the self-consistent Perron-Frobenius equation. Collective dynamics in globally coupled tent maps are re-examined, both with the help of direct simulation and the Perron-Frobenius equation. Collective chaos in a single...

As is well known, an irreducible nonnegative matrix possesses a uniquely determined Perron vector. As the main result of this paper we give a description of the set of Perron vectors of all the matrices contained in an irreducible nonnegative interval matrix A. This result is then applied to show that there exists a subset A∗ of A parameterized by n parameters (instead of n2 ones in the descrip...

The purpose of this paper is to present a unified Perron-Frobenius Theory for nonnegative, for real not necessarily nonnegative and for general complex matrices. The sign-real spectral radius was introduced for general real matrices. This quantity was shown to share certain properties with the Perron root of nonnegative matrices. In this paper we introduce the sign-complex spectral radius. Agai...

A matrix is said to have the Perron-Frobenius property (strong Perron-Frobenius property) if its spectral radius is an eigenvalue (a simple positive and strictly dominant eigenvalue) with a corresponding semipositive (positive) eigenvector. It is known that a matrix A with the Perron-Frobenius property can always be the limit of a sequence of matrices A(ε) with the strong Perron-Frobenius prope...

In this paper we consider a nonuniform unstability concept for evolution operators in Banach spaces. The relationship between this concept and the Perron condition is studied. Generalizations to the nonuniform case of some results of Van Minh, Räbiger and Schnaubelt are obtained. The theory we present here is applicable for general time varying linear equations in Banach spaces. 1. Evolution op...

We count various classes of algebraic integers of fixed degree by their largest absolute value. The classes of integers considered include all algebraic integers, Perron numbers, totally real integers, and totally complex integers. We give qualitative and quantitative results concerning the distribution of Perron numbers, answering in part a question of W. Thurston [Thu].