On general matrices having the Perron-Frobenius Property

Linear Algebra and its Applications, 2001

Linear Algebra and its Applications, 2004

In this paper, it is shown that the necessary and sufficient conditions on the Jordan form of a seminonnegative matrix, identified by Zaslavsky and McDonald, are in fact necessary and sufficient conditions on the Jordan form of every eventually nonnegative matrix. Thus every eventually nonnegative matrix is similar to a seminonnegative matrix. In Zaslavsky and McDonald, they show that several of the combinatorial properties of reducible nonnegative matrices carry over to reducible seminonnegative matrices. In this paper it is shown that a property on the index of cyclicity of an irreducible nonnegative matrix carries over to the seminonnegative matrices.

The celebrated Perron-Frobenius (PF) theorem is stated for irreducible nonnegative square matrices, and provides a simple characterization of their eigenvectors and eigenvalues. The importance of this theorem stems from the fact that eigenvalue problems on such matrices arise in many fields of science and engineering, including dynamical systems theory, economics, statistics and optimization. However, many real-life scenarios give rise to nonsquare matrices. Despite the extensive development of spectral theories for nonnegative matrices, the applicability of such theories to non-convex optimization problems is not clear. In particular, a natural question is whether the PF Theorem (along with its applications) can be generalized to a nonsquare setting. Our paper provides a generalization of the PF Theorem to nonsquare matrices. The extension can be interpreted as representing client-server systems with additional degrees of freedom, where each client may choose between multiple servers that can cooperate in serving it (while potentially interfering with other clients). This formulation is motivated by applications to power control in

2010

Matrix functions preserving several sets of generalized nonnegative matrices are characterized. These sets are PFn, the set of n × n real eventually positive matrices; and WPFn, the set of matrices A ∈ R n×n such that A and its transpose have the Perron-Frobenius property. Necessary and sufficient conditions for a matrix function to preserve the set of n × n real eventually nonnegative matrices and the set of n × n real exponentially nonnegative matrices are also presented. In particular, it is shown that if f (0) = 0 and f (0) = 0 for some entire function f , then such an entire function does not preserve the set of n × n real eventually nonnegative matrices. It is also shown that the only complex polynomials that preserve the set of n × n real exponentially nonnegative matrices are p(z) = az + b where a, b ∈ R and a ≥ 0.

2002

In this paper it is shown that an eventually nonnegative matrix A whose index of zero is less than or equal to one, exhibits many of the same combinatorial properties as a nonnegative matrix. In particular, there is a positive integer g such that Ag is nonnegative, A and Ag have the same irreducible classes, and the transitive closure of the reduced graph of A is the same as the transitive closure of the reduced graph of Ag. In this instance, many of the combinatorial properties of nonnegative matrices carry over to this subclass of the eventually nonnegative matrices. AMS subject classifications. 15A18, 15A48, 15A21

Linear Algebra and its Applications, 2003

In this paper we give necessary and sufficient conditions for a matrix in Jordan canonical form to be similar to an eventually nonnegative matrix whose irreducible diagonal blocks satisfy the conditions identified by Zaslavsky and Tam, and whose subdiagonal blocks (with respect to its Frobenius normal form) are nonnegative. These matrices are referred to as seminonnegative matrices, and we show that they exhibit many of the same combinatorial spectral properties as nonnegative matrices. This paper extends the work on Jordan forms of irreducible eventually nonnegative matrices to the reducible case.

Linear Algebra and its Applications

A 1. .. A n) i,j , where D ×D is the dimension of the matrices, U, V are respectively the largest entry and the smallest entry over all the positive entries of the matrices in A, and C is taken over all components in the dependency graph. The dependency graph is a directed graph where the vertices are the dimensions and there is an edge from i to j if and only if A i,j = 0 for some matrix A ∈ A. Furthermore, a bound on the norm is also given: There exists a nonnegative integer r so that for every n, const n r ρ(A) n ≤ max A1,...,An∈A A 1. .. A n ≤ const n r ρ(A) n. Corollaries of the approach include a simple proof for the joint spectral theorem for finite sets of nonnegative matrices and the convergence rate of some sequences. The method in use is mostly based on Fekete's lemma.

Linear Algebra and its Applications, 2010

We investigate (0, 1)-matrices which are totally nonnegative and therefore which have all of their eigenvalues equal to nonnegative real numbers. Such matrices are characterized by four forbidden submatrices (of orders 2 and 3). We show that the maximum number of 0s in an irreducible, totally nonnegative (0, 1)-matrix of order n is (n − 1) 2 and characterize those matrices with this number of 0s. We also show that the minimum Perron value of an irreducible, totally nonnegative (0, 1)-matrix of order n equals 2 + 2 cos 2π n+2 and characterize those matrices with this Perron value.

In this paper we give a sufficient condition for the existence and construction of a symmetric nonnegative matrix with prescribed spectrum, and a sufficient conditon for the existence and construction of a 4 × 4 symmetric nonnegative matrix with prescribed spectrum and diagonal entries. This last condition is independent of the sufficient condition given by Fiedler [LAA 9 (1974) 119-142]. We also give some partial answers on an open question of Guo [LAA 266 (1997) 261-270] about symmetric nonnegative matrices.

2013

We give a short proof of a recent result by Bernik, Mastnak, and Radjavi, stating that an irreducible group of complex matrices with nonnegative diagonal entries is diagonally similar to a group of nonnegative monomial matrices. We also explore the problem when an irreducible matrix semigroup in which each member is diagonally similar to a nonnegative matrix is diagonally similar to a semigroup of nonnegative matrices.