Related papers: The Perron-Frobenius Theorem for Multi-homogeneous…

The Perron-Frobenius theory for nonnegative matrices has been generalized to order-preserving homogeneous mappings on a cone and more recently to nonnegative multilinear forms. We unify both approaches by introducing the concept of…

We introduce the concept of shape partition of a tensor and formulate a general tensor eigenvalue problem that includes all previously studied eigenvalue problems as special cases. We formulate irreducibility and symmetry properties of a…

We prove an analog of Perron-Frobenius theorem for multilinear forms with nonnegative coefficients, and more generally, for polynomial maps with nonnegative coefficients. We determine the geometric convergence rate of the power algorithm to…

We study the l^{p_1,...,p_m} singular value problem for non-negative tensors. We prove a general Perron-Frobenius theorem for weakly irreducible and irreducible nonnegative tensors and provide a Collatz-Wielandt characterization of the…

The Perron-Frobenius theorem of nonnegative matrices is a classical result on spectral theory of matrices, which has wide applications in many domains. In this paper, we give the Perron-Frobenius theorem for dual tensors, that is, a dual…

If A is a nonnegative matrix whose associated directed graph is strongly connected, the Perron-Frobenius theorem asserts that A has an eigenvector in the positive cone, (R^+)^n. We associate a directed graph to any homogeneous, monotone…

Let $C$ be a closed cone with nonempty interior $C^\circ$ in a Banach space. Let $f:C^\circ \rightarrow C^\circ$ be an order-preserving subhomogeneous function with a fixed point in $C^\circ$. We introduce a condition which guarantees that…

For matrices with all nonnegative entries, the Perron-Frobenius theorem guarantees the existence of an eigenvector with all nonnegative components. We show that the existence of such an eigenvector is also guaranteed for a very different…

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 non-preserved cones. We introduce a new concept…

This paper studies the stability of discrete-time polynomial dynamical systems on hypergraphs by utilizing the Perron-Frobenius theorem for nonnegative tensors with respect to the tensors Z-eigenvalues and Z-eigenvectors. Firstly, for a…

The subleading eigenvalues and associated eigenfunctions of the Perron-Frobenius operator for 2-dimensional area-preserving maps are numerically investigated. We closely examine the validity of the so-called Ulam method, a numerical scheme…

We study the combinatorial and algebraic properties of Nonnegative Matrices. Our results are divided into three different categories. 1. We show a quantitative generalization of the 100 year-old Perron-Frobenius theorem, a fundamental…

The Perron-Frobenius theorem plays an important role in many areas of management science and operations research. This paper provides a probabilistic perspective on the theorem, by discussing a proof that exploits a probabilistic…

We prove that if an $n\times n$ matrix defined over ${\mathbb Q}_p$ (or more generally an arbitrary complete, discretely-valued, non-Archimedean field) satisfies a certain congruence property, then it has a strictly maximal eigenvalue in…

We propose a theory of eigenvalues, eigenvectors, singular values, and singular vectors for tensors based on a constrained variational approach much like the Rayleigh quotient for symmetric matrix eigenvalues. These notions are particularly…

We consider the transfer operators of non-uniformly expanding maps for potentials of various regularity, and show that a specific property of potentials ("flatness") implies a Ruelle-Perron-Frobenius Theorem and a decay of the transfer…

We establish a new functional central limit theorem result for non-invertible measure preserving maps that are not necessarily ergodic, using the Perron-Frobenius operator. We apply the result to asymptotically periodic transformations and…

The general solution of the inverse Frobenius-Perron problem considering the construction of a fully chaotic dynamical system with given invariant density is obtained within the class of one-dimensional unimodal maps. Some interesting…

Let $f:\mathbb{R}^n_{> 0} \rightarrow \mathbb{R}^n_{>0}$ be an order-preserving and homogeneous function. We show that the set of eigenvectors of $f$ in $\mathbb{R}^n_{>0}$ is nonempty and bounded in Hilbert's projective metric if and only…

We prove a random Ruelle--Perron--Frobenius theorem and the existence of relative equilibrium states for a class of random open and closed interval maps, without imposing transitivity requirements, such as mixing and covering conditions,…