NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS Numer. Linear Algebra Appl. 2009; ??:1–6 Prepared using nlaauth.cls [Version: 2002/09/18 v1.02] Two Characterizations of Matrices with the Perron-Frobenius Property Abed Elhashash and Daniel B. Szyld ∗ Department of Mathematics, Drexel University 3141 Chestnut Street, Philadelphia, Pennsylvania 19104-2816, USA (abed@drexel.edu) Department of Mathematics, Temple University (038-16) 1805 N. Broad Street, Philadelphia, Pennsylvania 19122-6094, USA (szyld@temple.edu) SUMMARY Two characterizations of general matrices for which the spectral radius is an eigenvalue and the corresponding eigenvector is either positive or nonnegative are presented. One is a full characterization in terms of the sign of the entries of the spectral projector. In another case, different necessary and sufficient conditions are presented which relate to the classes of the matrix. These characterizations generalize well-known results for nonnegative matrices. Copyright c  2009 John Wiley & Sons, Ltd. key words: Perron-Frobenius property; generalization of nonnegative matrices 1. INTRODUCTION We say that a real square matrix A has the Perron-Frobenius property if it has a right Perron- Frobenius eigenvector, i.e., if there exists a semipositive vector v for which Av = ρ(A)v, where ρ(A) stands for the spectral radius of A. A semipositive vector is a nonzero nonnegative vector (see, e.g. [1]), where here and throughout the paper, the nonnegativity or positivity of vectors and matrices, and inequalities involving them, are understood to be componentwise. A left Perron-Frobenius eigenvector of A is the right Perron-Frobenius eigenvector of A T , the transpose of A. Often the right (or left) Perron-Frobenius eigenvector is called a Perron eigenvector, or simply a Perron vector. Matrices with the Perron-Frobenius property include positive matrices, and irreducible nonnegative matrices. In these cases the Perron eigenvector is positive and the spectral radius is a strictly dominant eigenvalue, i.e., the spectral radius is the only eigenvalue with the largest modulus. Nonnegative matrices, not necessarily irreducible, also have the Perron- Frobenius property. This follows from the Perron-Frobenius theorem [8], [18]. This theorem * Correspondence to: Daniel B. Szyld, Department of Mathematics, Temple University (038-16) 1805 N. Broad Street, Philadelphia, Pennsylvania 19122-6094, USA (szyld@temple.edu) Contract/grant sponsor: U.S. Department of Energy; contract/grant number: DE-FG02-05ER25672 Received 13 August 2008 Copyright c  2009 John Wiley & Sons, Ltd. Revised 6 February 2009