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Abstract A new upper bound for the spectral radius of matrices having the Perron–Frobenius property is given by considering the position of positive entries. Some examples involving the largest zero of polynomials and the spectral radius of the iterative matrix for the Perron–Frobenius splitting are given to show the superiority of the theoretical result.

A new upper bound for the spectral radius of matrices having the Perron–Frobenius property is given by considering the position of positive entries. Some examples involving the largest zero of polynomials and the spectral radius of the iterative matrix for the Perron–Frobenius splitting are given to show the superiority of the theoretical result. [ABSTRACT FROM AUTHOR]

We in this paper define the outer-Perron-Frobenius splitting, which is an extension of the pseudo- Perron-Frobenius splitting defined in [A.N. Sushama, K. Premakumari, K.C. Sivakumar, Extensions of Perron-Frobenius splittings and relationships with nonnegative Moore-Penrose inverse, Linear and Multilinear Algebra 63 (2015) 1-11]. We present some criteria for the convergence of the outer-Perron-Frobenius splitting. The findings of this paper generalize some known results in the literatures.

A matrix A ∈ ℝn×n is a GM-matrix if A = sI − B, where 0 < ρ(B) ≤ s and B ∈WPFn i.e., both B and Bthave ρ(B) as their eigenvalues and their corresponding eigenvector is entry wise nonnegative. In this article,we consider a generalization of a subclass of GM-matrices having a nonnegative core nilpotent decompositionand prove a characterization result for such matrices. Also, we study various notions of splitting of matricesfrom this new class and obtain sufficient conditions for their convergence.

An-matrix has the form, whereandis eventually nonnegative, i.e.is entry wise nonnegative for all sufficiently large integers. In this article, two new types of splittings of matrices are introduced. The class of matrices possessing a splitting of one of these types includes-matrices as a subclass. The authors derived necessary and sufficient conditions for the convergence of these splittings in terms of an extended notion of the nonnegativity of the Moore–Penrose inverse. The work reported here widens the applicability of the existing results. [ABSTRACT FROM AUTHOR]

Abstract: We extend the theory of nonnegative matrices to the matrices that have some negative entries. We present and prove some properties which give us information, when a matrix possesses a Perron–Frobenius eigenpair. We apply also this theory by proposing the Perron–Frobenius splitting for the solution of the linear system Ax = b by classical iterative methods. Perron–Frobenius splittings constitute an extension of the well known regular splittings, weak regular splittings and nonnegative splittings. Convergence and comparison properties are given and proved. [Copyright &y& Elsevier]